:: FUNCT_3 semantic presentation
theorem Th1: :: FUNCT_3:1
for b
1, b
2 being
set holds
( b
1 c= b
2 implies
id b
1 = (id b2) | b
1 )
theorem Th2: :: FUNCT_3:2
theorem Th3: :: FUNCT_3:3
theorem Th4: :: FUNCT_3:4
theorem Th5: :: FUNCT_3:5
scheme :: FUNCT_3:sch 1
s1{ F
1()
-> set , F
2()
-> set , P
1[
set ,
set ,
set ] } :
ex b
1 being
Function st
(
dom b
1 = [:F1(),F2():] & ( for b
2, b
3 being
set holds
( b
2 in F
1() & b
3 in F
2() implies P
1[b
2,b
3,b
1 . [b2,b3]] ) ) )
provided
E6:
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
1[b
1,b
2,b
3] & P
1[b
1,b
2,b
4] implies b
3 = b
4 )
and
E7:
for b
1, b
2 being
set holds
not ( b
1 in F
1() & b
2 in F
2() & ( for b
3 being
set holds
not P
1[b
1,b
2,b
3] ) )
theorem Th6: :: FUNCT_3:6
:: deftheorem Def1 defines .: FUNCT_3:def 1 :
theorem Th7: :: FUNCT_3:7
canceled;
theorem Th8: :: FUNCT_3:8
theorem Th9: :: FUNCT_3:9
theorem Th10: :: FUNCT_3:10
theorem Th11: :: FUNCT_3:11
canceled;
theorem Th12: :: FUNCT_3:12
theorem Th13: :: FUNCT_3:13
theorem Th14: :: FUNCT_3:14
theorem Th15: :: FUNCT_3:15
theorem Th16: :: FUNCT_3:16
theorem Th17: :: FUNCT_3:17
theorem Th18: :: FUNCT_3:18
theorem Th19: :: FUNCT_3:19
theorem Th20: :: FUNCT_3:20
theorem Th21: :: FUNCT_3:21
theorem Th22: :: FUNCT_3:22
:: deftheorem Def2 defines " FUNCT_3:def 2 :
theorem Th23: :: FUNCT_3:23
canceled;
theorem Th24: :: FUNCT_3:24
theorem Th25: :: FUNCT_3:25
theorem Th26: :: FUNCT_3:26
canceled;
theorem Th27: :: FUNCT_3:27
theorem Th28: :: FUNCT_3:28
theorem Th29: :: FUNCT_3:29
theorem Th30: :: FUNCT_3:30
theorem Th31: :: FUNCT_3:31
theorem Th32: :: FUNCT_3:32
theorem Th33: :: FUNCT_3:33
theorem Th34: :: FUNCT_3:34
theorem Th35: :: FUNCT_3:35
theorem Th36: :: FUNCT_3:36
theorem Th37: :: FUNCT_3:37
theorem Th38: :: FUNCT_3:38
theorem Th39: :: FUNCT_3:39
definition
let c
1, c
2 be
set ;
func chi c
1,c
2 -> Function means :
Def3:
:: FUNCT_3:def 3
(
dom a
3 = a
2 & ( for b
1 being
set holds
( b
1 in a
2 implies ( ( b
1 in a
1 implies a
3 . b
1 = 1 ) & ( not b
1 in a
1 implies a
3 . b
1 = 0 ) ) ) ) );
existence
ex b1 being Function st
( dom b1 = c2 & ( for b2 being set holds
( b2 in c2 implies ( ( b2 in c1 implies b1 . b2 = 1 ) & ( not b2 in c1 implies b1 . b2 = 0 ) ) ) ) )
uniqueness
for b1, b2 being Function holds
( dom b1 = c2 & ( for b3 being set holds
( b3 in c2 implies ( ( b3 in c1 implies b1 . b3 = 1 ) & ( not b3 in c1 implies b1 . b3 = 0 ) ) ) ) & dom b2 = c2 & ( for b3 being set holds
( b3 in c2 implies ( ( b3 in c1 implies b2 . b3 = 1 ) & ( not b3 in c1 implies b2 . b3 = 0 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines chi FUNCT_3:def 3 :
for b
1, b
2 being
set for b
3 being
Function holds
( b
3 = chi b
1,b
2 iff (
dom b
3 = b
2 & ( for b
4 being
set holds
( b
4 in b
2 implies ( ( b
4 in b
1 implies b
3 . b
4 = 1 ) & ( not b
4 in b
1 implies b
3 . b
4 = 0 ) ) ) ) ) );
theorem Th40: :: FUNCT_3:40
canceled;
theorem Th41: :: FUNCT_3:41
canceled;
theorem Th42: :: FUNCT_3:42
for b
1, b
2, b
3 being
set holds
(
(chi b2,b3) . b
1 = 1 implies b
1 in b
2 )
theorem Th43: :: FUNCT_3:43
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 \ b
3 implies
(chi b3,b2) . b
1 = 0 )
theorem Th44: :: FUNCT_3:44
canceled;
theorem Th45: :: FUNCT_3:45
canceled;
theorem Th46: :: FUNCT_3:46
canceled;
theorem Th47: :: FUNCT_3:47
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
3 c= b
2 &
chi b
1,b
2 = chi b
3,b
2 implies b
1 = b
3 )
theorem Th48: :: FUNCT_3:48
theorem Th49: :: FUNCT_3:49
theorem Th50: :: FUNCT_3:50
canceled;
theorem Th51: :: FUNCT_3:51
canceled;
theorem Th52: :: FUNCT_3:52
canceled;
theorem Th53: :: FUNCT_3:53
theorem Th54: :: FUNCT_3:54
canceled;
theorem Th55: :: FUNCT_3:55
canceled;
theorem Th56: :: FUNCT_3:56
for b
1, b
2 being
set for b
3 being
Subset of b
2 holds
( b
1 in b
3 implies
(incl b3) . b
1 in b
2 )
definition
let c
1, c
2 be
set ;
canceled;func pr1 c
1,c
2 -> Function means :
Def5:
:: FUNCT_3:def 5
(
dom a
3 = [:a1,a2:] & ( for b
1, b
2 being
set holds
( b
1 in a
1 & b
2 in a
2 implies a
3 . [b1,b2] = b
1 ) ) );
existence
ex b1 being Function st
( dom b1 = [:c1,c2:] & ( for b2, b3 being set holds
( b2 in c1 & b3 in c2 implies b1 . [b2,b3] = b2 ) ) )
uniqueness
for b1, b2 being Function holds
( dom b1 = [:c1,c2:] & ( for b3, b4 being set holds
( b3 in c1 & b4 in c2 implies b1 . [b3,b4] = b3 ) ) & dom b2 = [:c1,c2:] & ( for b3, b4 being set holds
( b3 in c1 & b4 in c2 implies b2 . [b3,b4] = b3 ) ) implies b1 = b2 )
func pr2 c
1,c
2 -> Function means :
Def6:
:: FUNCT_3:def 6
(
dom a
3 = [:a1,a2:] & ( for b
1, b
2 being
set holds
( b
1 in a
1 & b
2 in a
2 implies a
3 . [b1,b2] = b
2 ) ) );
existence
ex b1 being Function st
( dom b1 = [:c1,c2:] & ( for b2, b3 being set holds
( b2 in c1 & b3 in c2 implies b1 . [b2,b3] = b3 ) ) )
uniqueness
for b1, b2 being Function holds
( dom b1 = [:c1,c2:] & ( for b3, b4 being set holds
( b3 in c1 & b4 in c2 implies b1 . [b3,b4] = b4 ) ) & dom b2 = [:c1,c2:] & ( for b3, b4 being set holds
( b3 in c1 & b4 in c2 implies b2 . [b3,b4] = b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def4 FUNCT_3:def 4 :
canceled;
:: deftheorem Def5 defines pr1 FUNCT_3:def 5 :
for b
1, b
2 being
set for b
3 being
Function holds
( b
3 = pr1 b
1,b
2 iff (
dom b
3 = [:b1,b2:] & ( for b
4, b
5 being
set holds
( b
4 in b
1 & b
5 in b
2 implies b
3 . [b4,b5] = b
4 ) ) ) );
:: deftheorem Def6 defines pr2 FUNCT_3:def 6 :
for b
1, b
2 being
set for b
3 being
Function holds
( b
3 = pr2 b
1,b
2 iff (
dom b
3 = [:b1,b2:] & ( for b
4, b
5 being
set holds
( b
4 in b
1 & b
5 in b
2 implies b
3 . [b4,b5] = b
5 ) ) ) );
theorem Th57: :: FUNCT_3:57
canceled;
theorem Th58: :: FUNCT_3:58
canceled;
theorem Th59: :: FUNCT_3:59
theorem Th60: :: FUNCT_3:60
theorem Th61: :: FUNCT_3:61
theorem Th62: :: FUNCT_3:62
definition
let c
1, c
2 be
set ;
redefine func pr1 as
pr1 c
1,c
2 -> Function of
[:a1,a2:],a
1;
coherence
pr1 c1,c2 is Function of [:c1,c2:],c1
redefine func pr2 as
pr2 c
1,c
2 -> Function of
[:a1,a2:],a
2;
coherence
pr2 c1,c2 is Function of [:c1,c2:],c2
end;
:: deftheorem Def7 defines delta FUNCT_3:def 7 :
for b
1 being
set for b
2 being
Function holds
( b
2 = delta b
1 iff (
dom b
2 = b
1 & ( for b
3 being
set holds
( b
3 in b
1 implies b
2 . b
3 = [b3,b3] ) ) ) );
theorem Th63: :: FUNCT_3:63
canceled;
theorem Th64: :: FUNCT_3:64
canceled;
theorem Th65: :: FUNCT_3:65
canceled;
theorem Th66: :: FUNCT_3:66
:: deftheorem Def8 defines <: FUNCT_3:def 8 :
theorem Th67: :: FUNCT_3:67
canceled;
theorem Th68: :: FUNCT_3:68
theorem Th69: :: FUNCT_3:69
theorem Th70: :: FUNCT_3:70
theorem Th71: :: FUNCT_3:71
theorem Th72: :: FUNCT_3:72
theorem Th73: :: FUNCT_3:73
theorem Th74: :: FUNCT_3:74
theorem Th75: :: FUNCT_3:75
theorem Th76: :: FUNCT_3:76
theorem Th77: :: FUNCT_3:77
theorem Th78: :: FUNCT_3:78
theorem Th79: :: FUNCT_3:79
theorem Th80: :: FUNCT_3:80
theorem Th81: :: FUNCT_3:81
theorem Th82: :: FUNCT_3:82
theorem Th83: :: FUNCT_3:83
for b
1, b
2, b
3 being
set for b
4, b
5 being
Function of b
1,b
2for b
6, b
7 being
Function of b
1,b
3 holds
( ( b
2 = {} implies b
1 = {} ) & ( b
3 = {} implies b
1 = {} ) &
<:b4,b6:> = <:b5,b7:> implies ( b
4 = b
5 & b
6 = b
7 ) )
theorem Th84: :: FUNCT_3:84
definition
let c
1, c
2 be
Function;
func [:c1,c2:] -> Function means :
Def9:
:: FUNCT_3:def 9
(
dom a
3 = [:(dom a1),(dom a2):] & ( for b
1, b
2 being
set holds
( b
1 in dom a
1 & b
2 in dom a
2 implies a
3 . [b1,b2] = [(a1 . b1),(a2 . b2)] ) ) );
existence
ex b1 being Function st
( dom b1 = [:(dom c1),(dom c2):] & ( for b2, b3 being set holds
( b2 in dom c1 & b3 in dom c2 implies b1 . [b2,b3] = [(c1 . b2),(c2 . b3)] ) ) )
uniqueness
for b1, b2 being Function holds
( dom b1 = [:(dom c1),(dom c2):] & ( for b3, b4 being set holds
( b3 in dom c1 & b4 in dom c2 implies b1 . [b3,b4] = [(c1 . b3),(c2 . b4)] ) ) & dom b2 = [:(dom c1),(dom c2):] & ( for b3, b4 being set holds
( b3 in dom c1 & b4 in dom c2 implies b2 . [b3,b4] = [(c1 . b3),(c2 . b4)] ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines [: FUNCT_3:def 9 :
theorem Th85: :: FUNCT_3:85
canceled;
theorem Th86: :: FUNCT_3:86
theorem Th87: :: FUNCT_3:87
theorem Th88: :: FUNCT_3:88
theorem Th89: :: FUNCT_3:89
theorem Th90: :: FUNCT_3:90
theorem Th91: :: FUNCT_3:91
theorem Th92: :: FUNCT_3:92
theorem Th93: :: FUNCT_3:93
theorem Th94: :: FUNCT_3:94
theorem Th95: :: FUNCT_3:95
definition
let c
1, c
2, c
3, c
4 be
set ;
let c
5 be
Function of c
1,c
3;
let c
6 be
Function of c
2,c
4;
redefine func [: as
[:c5,c6:] -> Function of
[:a1,a2:],
[:a3,a4:];
coherence
[:c5,c6:] is Function of [:c1,c2:],[:c3,c4:]
by Th95;
end;
theorem Th96: :: FUNCT_3:96
theorem Th97: :: FUNCT_3:97
theorem Th98: :: FUNCT_3:98
theorem Th99: :: FUNCT_3:99
theorem Th100: :: FUNCT_3:100