:: FUNCT_2 semantic presentation
:: deftheorem Def1 defines quasi_total FUNCT_2:def 1 :
theorem Th1: :: FUNCT_2:1
canceled;
theorem Th2: :: FUNCT_2:2
canceled;
theorem Th3: :: FUNCT_2:3
theorem Th4: :: FUNCT_2:4
theorem Th5: :: FUNCT_2:5
for b
1, b
2 being
set for b
3 being
Function holds
(
dom b
3 = b
1 & ( for b
4 being
set holds
( b
4 in b
1 implies b
3 . b
4 in b
2 ) ) implies b
3 is
Function of b
1,b
2 )
theorem Th6: :: FUNCT_2:6
theorem Th7: :: FUNCT_2:7
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( b
2 <> {} & b
3 in b
1 implies b
4 . b
3 in b
2 )
theorem Th8: :: FUNCT_2:8
theorem Th9: :: FUNCT_2:9
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) & b
2 c= b
3 implies b
4 is
Function of b
1,b
3 )
:: deftheorem Def2 defines Funcs FUNCT_2:def 2 :
for b
1, b
2, b
3 being
set holds
( b
3 = Funcs b
1,b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
Function st
( b
4 = b
5 &
dom b
5 = b
1 &
rng b
5 c= b
2 ) ) );
theorem Th10: :: FUNCT_2:10
canceled;
theorem Th11: :: FUNCT_2:11
theorem Th12: :: FUNCT_2:12
theorem Th13: :: FUNCT_2:13
canceled;
theorem Th14: :: FUNCT_2:14
theorem Th15: :: FUNCT_2:15
canceled;
theorem Th16: :: FUNCT_2:16
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( b
2 <> {} & ( for b
4 being
set holds
not ( b
4 in b
2 & ( for b
5 being
set holds
not ( b
5 in b
1 & b
4 = b
3 . b
5 ) ) ) ) implies
rng b
3 = b
2 )
theorem Th17: :: FUNCT_2:17
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
not ( b
3 in rng b
4 & ( for b
5 being
set holds
not ( b
5 in b
1 & b
4 . b
5 = b
3 ) ) )
theorem Th18: :: FUNCT_2:18
for b
1, b
2 being
set for b
3, b
4 being
Function of b
1,b
2 holds
( ( for b
5 being
set holds
( b
5 in b
1 implies b
3 . b
5 = b
4 . b
5 ) ) implies b
3 = b
4 )
theorem Th19: :: FUNCT_2:19
theorem Th20: :: FUNCT_2:20
theorem Th21: :: FUNCT_2:21
theorem Th22: :: FUNCT_2:22
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( b
2 <> {} implies (
rng b
3 = b
2 iff for b
4 being
set holds
( b
4 <> {} implies for b
5, b
6 being
Function of b
2,b
4 holds
( b
5 * b
3 = b
6 * b
3 implies b
5 = b
6 ) ) ) )
theorem Th23: :: FUNCT_2:23
for b
1, b
2 being
set for b
3 being
Relation of b
1,b
2 holds
(
(id b1) * b
3 = b
3 & b
3 * (id b2) = b
3 )
theorem Th24: :: FUNCT_2:24
theorem Th25: :: FUNCT_2:25
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) implies ( b
3 is
one-to-one iff for b
4, b
5 being
set holds
( b
4 in b
1 & b
5 in b
1 & b
3 . b
4 = b
3 . b
5 implies b
4 = b
5 ) ) )
theorem Th26: :: FUNCT_2:26
theorem Th27: :: FUNCT_2:27
theorem Th28: :: FUNCT_2:28
theorem Th29: :: FUNCT_2:29
theorem Th30: :: FUNCT_2:30
theorem Th31: :: FUNCT_2:31
theorem Th32: :: FUNCT_2:32
theorem Th33: :: FUNCT_2:33
canceled;
theorem Th34: :: FUNCT_2:34
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2for b
4 being
Function of b
2,b
1 holds
( b
1 <> {} & b
2 <> {} &
rng b
3 = b
2 & b
3 is
one-to-one & ( for b
5, b
6 being
set holds
( ( b
5 in b
2 & b
4 . b
5 = b
6 implies ( b
6 in b
1 & b
3 . b
6 = b
5 ) ) & ( b
6 in b
1 & b
3 . b
6 = b
5 implies ( b
5 in b
2 & b
4 . b
5 = b
6 ) ) ) ) implies b
4 = b
3 " )
theorem Th35: :: FUNCT_2:35
theorem Th36: :: FUNCT_2:36
theorem Th37: :: FUNCT_2:37
theorem Th38: :: FUNCT_2:38
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) & b
3 c= b
1 implies b
4 | b
3 is
Function of b
3,b
2 )
theorem Th39: :: FUNCT_2:39
canceled;
theorem Th40: :: FUNCT_2:40
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( b
1 c= b
3 implies b
4 | b
3 = b
4 )
theorem Th41: :: FUNCT_2:41
for b
1, b
2, b
3, b
4 being
set for b
5 being
Function of b
1,b
2 holds
( b
2 <> {} & b
3 in b
1 & b
5 . b
3 in b
4 implies
(b4 | b5) . b
3 = b
5 . b
3 )
theorem Th42: :: FUNCT_2:42
canceled;
theorem Th43: :: FUNCT_2:43
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( b
2 <> {} implies for b
5 being
set holds
( ex b
6 being
set st
( b
6 in b
1 & b
6 in b
3 & b
5 = b
4 . b
6 ) implies b
5 in b
4 .: b
3 ) )
theorem Th44: :: FUNCT_2:44
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds b
4 .: b
3 c= b
2 ;
theorem Th45: :: FUNCT_2:45
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) implies b
3 .: b
1 = rng b
3 )
theorem Th46: :: FUNCT_2:46
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( b
2 <> {} implies for b
5 being
set holds
( b
5 in b
4 " b
3 iff ( b
5 in b
1 & b
4 . b
5 in b
3 ) ) )
theorem Th47: :: FUNCT_2:47
for b
1, b
2, b
3 being
set for b
4 being
PartFunc of b
1,b
2 holds b
4 " b
3 c= b
1 ;
theorem Th48: :: FUNCT_2:48
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) implies b
3 " b
2 = b
1 )
theorem Th49: :: FUNCT_2:49
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( ( for b
4 being
set holds
not ( b
4 in b
2 & not b
3 " {b4} <> {} ) ) iff
rng b
3 = b
2 )
theorem Th50: :: FUNCT_2:50
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) & b
3 c= b
1 implies b
3 c= b
4 " (b4 .: b3) )
theorem Th51: :: FUNCT_2:51
for b
1, b
2 being
set for b
3 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) implies b
3 " (b3 .: b1) = b
1 )
theorem Th52: :: FUNCT_2:52
canceled;
theorem Th53: :: FUNCT_2:53
for b
1, b
2, b
3, b
4 being
set for b
5 being
Function of b
1,b
2for b
6 being
Function of b
2,b
3 holds
( ( b
3 = {} implies b
2 = {} ) & ( b
2 = {} implies b
1 = {} ) implies b
5 " b
4 c= (b6 * b5) " (b6 .: b4) )
theorem Th54: :: FUNCT_2:54
canceled;
theorem Th55: :: FUNCT_2:55
theorem Th56: :: FUNCT_2:56
canceled;
theorem Th57: :: FUNCT_2:57
canceled;
theorem Th58: :: FUNCT_2:58
canceled;
theorem Th59: :: FUNCT_2:59
theorem Th60: :: FUNCT_2:60
theorem Th61: :: FUNCT_2:61
theorem Th62: :: FUNCT_2:62
theorem Th63: :: FUNCT_2:63
canceled;
theorem Th64: :: FUNCT_2:64
theorem Th65: :: FUNCT_2:65
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,
{b2} holds
( b
3 in b
1 implies b
4 . b
3 = b
2 )
theorem Th66: :: FUNCT_2:66
for b
1, b
2 being
set for b
3, b
4 being
Function of b
1,
{b2} holds b
3 = b
4
theorem Th67: :: FUNCT_2:67
theorem Th68: :: FUNCT_2:68
canceled;
theorem Th69: :: FUNCT_2:69
canceled;
theorem Th70: :: FUNCT_2:70
for b
1, b
2 being
set for b
3 being
Function of b
1,b
1for b
4 being
Function holds
( b
2 in b
1 implies
(b4 * b3) . b
2 = b
4 . (b3 . b2) )
theorem Th71: :: FUNCT_2:71
canceled;
theorem Th72: :: FUNCT_2:72
canceled;
theorem Th73: :: FUNCT_2:73
theorem Th74: :: FUNCT_2:74
canceled;
theorem Th75: :: FUNCT_2:75
for b
1 being
set for b
2, b
3 being
Function of b
1,b
1 holds
( b
3 * b
2 = b
2 &
rng b
2 = b
1 implies b
3 = id b
1 )
theorem Th76: :: FUNCT_2:76
theorem Th77: :: FUNCT_2:77
for b
1 being
set for b
2 being
Function of b
1,b
1 holds
( b
2 is
one-to-one iff for b
3, b
4 being
set holds
( b
3 in b
1 & b
4 in b
1 & b
2 . b
3 = b
2 . b
4 implies b
3 = b
4 ) )
theorem Th78: :: FUNCT_2:78
canceled;
theorem Th79: :: FUNCT_2:79
theorem Th80: :: FUNCT_2:80
canceled;
theorem Th81: :: FUNCT_2:81
canceled;
theorem Th82: :: FUNCT_2:82
:: deftheorem Def3 defines onto FUNCT_2:def 3 :
:: deftheorem Def4 defines bijective FUNCT_2:def 4 :
theorem Th83: :: FUNCT_2:83
theorem Th84: :: FUNCT_2:84
canceled;
theorem Th85: :: FUNCT_2:85
theorem Th86: :: FUNCT_2:86
theorem Th87: :: FUNCT_2:87
theorem Th88: :: FUNCT_2:88
theorem Th89: :: FUNCT_2:89
canceled;
theorem Th90: :: FUNCT_2:90
canceled;
theorem Th91: :: FUNCT_2:91
canceled;
theorem Th92: :: FUNCT_2:92
for b
1, b
2 being
set for b
3 being
Permutation of b
1 holds
( b
2 c= b
1 implies ( b
3 .: (b3 " b2) = b
2 & b
3 " (b3 .: b2) = b
2 ) )
theorem Th93: :: FUNCT_2:93
canceled;
theorem Th94: :: FUNCT_2:94
canceled;
theorem Th95: :: FUNCT_2:95
canceled;
theorem Th96: :: FUNCT_2:96
canceled;
theorem Th97: :: FUNCT_2:97
canceled;
theorem Th98: :: FUNCT_2:98
canceled;
theorem Th99: :: FUNCT_2:99
canceled;
theorem Th100: :: FUNCT_2:100
canceled;
theorem Th101: :: FUNCT_2:101
canceled;
theorem Th102: :: FUNCT_2:102
canceled;
theorem Th103: :: FUNCT_2:103
canceled;
theorem Th104: :: FUNCT_2:104
canceled;
theorem Th105: :: FUNCT_2:105
canceled;
theorem Th106: :: FUNCT_2:106
canceled;
theorem Th107: :: FUNCT_2:107
canceled;
theorem Th108: :: FUNCT_2:108
canceled;
theorem Th109: :: FUNCT_2:109
canceled;
theorem Th110: :: FUNCT_2:110
canceled;
theorem Th111: :: FUNCT_2:111
canceled;
theorem Th112: :: FUNCT_2:112
canceled;
theorem Th113: :: FUNCT_2:113
for b
1, b
2 being
set for b
3, b
4 being
Function of b
1,b
2 holds
( ( for b
5 being
Element of b
1 holds b
3 . b
5 = b
4 . b
5 ) implies b
3 = b
4 )
theorem Th114: :: FUNCT_2:114
canceled;
theorem Th115: :: FUNCT_2:115
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2for b
5 being
set holds
not ( b
5 in b
4 .: b
3 & ( for b
6 being
set holds
not ( b
6 in b
1 & b
6 in b
3 & b
5 = b
4 . b
6 ) ) )
theorem Th116: :: FUNCT_2:116
for b
1, b
2, b
3 being
set for b
4 being
Function of b
1,b
2for b
5 being
set holds
not ( b
5 in b
4 .: b
3 & ( for b
6 being
Element of b
1 holds
not ( b
6 in b
3 & b
5 = b
4 . b
6 ) ) )
theorem Th117: :: FUNCT_2:117
canceled;
theorem Th118: :: FUNCT_2:118
canceled;
theorem Th119: :: FUNCT_2:119
canceled;
theorem Th120: :: FUNCT_2:120
canceled;
theorem Th121: :: FUNCT_2:121
scheme :: FUNCT_2:sch 5
s5{ F
1()
-> set , F
2()
-> set , P
1[
set ], F
3(
set )
-> set , F
4(
set )
-> set } :
ex b
1 being
Function of F
1(),F
2() st
for b
2 being
set holds
( b
2 in F
1() implies ( ( P
1[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( not P
1[b
2] implies b
1 . b
2 = F
4(b
2) ) ) )
provided
E27:
for b
1 being
set holds
( b
1 in F
1() implies ( ( P
1[b
1] implies F
3(b
1)
in F
2() ) & ( not P
1[b
1] implies F
4(b
1)
in F
2() ) ) )
theorem Th122: :: FUNCT_2:122
canceled;
theorem Th123: :: FUNCT_2:123
canceled;
theorem Th124: :: FUNCT_2:124
canceled;
theorem Th125: :: FUNCT_2:125
canceled;
theorem Th126: :: FUNCT_2:126
canceled;
theorem Th127: :: FUNCT_2:127
canceled;
theorem Th128: :: FUNCT_2:128
canceled;
theorem Th129: :: FUNCT_2:129
canceled;
theorem Th130: :: FUNCT_2:130
theorem Th131: :: FUNCT_2:131
theorem Th132: :: FUNCT_2:132
theorem Th133: :: FUNCT_2:133
theorem Th134: :: FUNCT_2:134
theorem Th135: :: FUNCT_2:135
canceled;
theorem Th136: :: FUNCT_2:136
for b
1, b
2 being
set for b
3 being
PartFunc of b
1,b
2 holds
not ( ( b
2 = {} implies b
1 = {} ) & ( for b
4 being
Function of b
1,b
2 holds
ex b
5 being
set st
( b
5 in dom b
3 & not b
4 . b
5 = b
3 . b
5 ) ) )
theorem Th137: :: FUNCT_2:137
canceled;
theorem Th138: :: FUNCT_2:138
canceled;
theorem Th139: :: FUNCT_2:139
canceled;
theorem Th140: :: FUNCT_2:140
canceled;
theorem Th141: :: FUNCT_2:141
theorem Th142: :: FUNCT_2:142
theorem Th143: :: FUNCT_2:143
theorem Th144: :: FUNCT_2:144
canceled;
theorem Th145: :: FUNCT_2:145
theorem Th146: :: FUNCT_2:146
theorem Th147: :: FUNCT_2:147
canceled;
theorem Th148: :: FUNCT_2:148
theorem Th149: :: FUNCT_2:149
theorem Th150: :: FUNCT_2:150
canceled;
theorem Th151: :: FUNCT_2:151
theorem Th152: :: FUNCT_2:152
theorem Th153: :: FUNCT_2:153
canceled;
theorem Th154: :: FUNCT_2:154
theorem Th155: :: FUNCT_2:155
theorem Th156: :: FUNCT_2:156
theorem Th157: :: FUNCT_2:157
canceled;
theorem Th158: :: FUNCT_2:158
theorem Th159: :: FUNCT_2:159
theorem Th160: :: FUNCT_2:160
theorem Th161: :: FUNCT_2:161
theorem Th162: :: FUNCT_2:162
theorem Th163: :: FUNCT_2:163
canceled;
theorem Th164: :: FUNCT_2:164
theorem Th165: :: FUNCT_2:165
theorem Th166: :: FUNCT_2:166
theorem Th167: :: FUNCT_2:167
theorem Th168: :: FUNCT_2:168
definition
let c
1, c
2, c
3 be
set ;
func c
1,c
2 :-> c
3 -> Function of
[:{a1},{a2}:],
{a3} means :: FUNCT_2:def 5
verum;
existence
ex b1 being Function of [:{c1},{c2}:],{c3} st
verum
;
uniqueness
for b1, b2 being Function of [:{c1},{c2}:],{c3} holds b1 = b2
by Th66;
end;
:: deftheorem Def5 defines :-> FUNCT_2:def 5 :
theorem Th169: :: FUNCT_2:169
definition
let c
1, c
2, c
3 be non
empty set ;
let c
4 be
Function of c
1,
[:c2,c3:];
redefine func pr1 as
pr1 c
4 -> Function of a
1,a
2 means :: FUNCT_2:def 6
for b
1 being
Element of a
1 holds a
5 . b
1 = (a4 . b1) `1 ;
coherence
pr1 c4 is Function of c1,c2
compatibility
for b1 being Function of c1,c2 holds
( b1 = pr1 c4 iff for b2 being Element of c1 holds b1 . b2 = (c4 . b2) `1 )
redefine func pr2 as
pr2 c
4 -> Function of a
1,a
3 means :: FUNCT_2:def 7
for b
1 being
Element of a
1 holds a
5 . b
1 = (a4 . b1) `2 ;
coherence
pr2 c4 is Function of c1,c3
compatibility
for b1 being Function of c1,c3 holds
( b1 = pr2 c4 iff for b2 being Element of c1 holds b1 . b2 = (c4 . b2) `2 )
end;
:: deftheorem Def6 defines pr1 FUNCT_2:def 6 :
:: deftheorem Def7 defines pr2 FUNCT_2:def 7 :
:: deftheorem Def8 defines = FUNCT_2:def 8 :
:: deftheorem Def9 defines = FUNCT_2:def 9 :
for b
1 being
set for b
2 being non
empty set for b
3, b
4 being
Function of b
1,b
2 holds
( b
3 = b
4 iff for b
5 being
Element of b
1 holds b
3 . b
5 = b
4 . b
5 );