:: ZF_FUND1 semantic presentation
definition
let c
1, c
2 be
set ;
func c
1 (#) c
2 -> set means :
Def1:
:: ZF_FUND1:def 1
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3, b
4 being
set st
( b
1 = [b2,b4] &
[b2,b3] in a
1 &
[b3,b4] in a
2 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4, b5 being set st
( b2 = [b3,b5] & [b3,b4] in c1 & [b4,b5] in c2 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5, b6 being set st
( b3 = [b4,b6] & [b4,b5] in c1 & [b5,b6] in c2 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5, b6 being set st
( b3 = [b4,b6] & [b4,b5] in c1 & [b5,b6] in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines (#) ZF_FUND1:def 1 :
for b
1, b
2, b
3 being
set holds
( b
3 = b
1 (#) b
2 iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5, b
6, b
7 being
set st
( b
4 = [b5,b7] &
[b5,b6] in b
1 &
[b6,b7] in b
2 ) ) );
:: deftheorem Def2 defines decode ZF_FUND1:def 2 :
:: deftheorem Def3 defines x". ZF_FUND1:def 3 :
Lemma4:
( dom decode = omega & rng decode = VAR & decode is one-to-one & decode " is one-to-one & dom (decode " ) = VAR & rng (decode " ) = omega )
:: deftheorem Def4 defines code ZF_FUND1:def 4 :
definition
let c
1 be
ZF-formula;
let c
2 be non
empty set ;
func Diagram c
1,c
2 -> set means :
Def5:
:: ZF_FUND1:def 5
for b
1 being
set holds
( b
1 in a
3 iff ex b
2 being
Function of
VAR ,a
2 st
( b
1 = (b2 * decode ) | (code (Free a1)) & b
2 in St a
1,a
2 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3 being Function of VAR ,c2 st
( b2 = (b3 * decode ) | (code (Free c1)) & b3 in St c1,c2 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4 being Function of VAR ,c2 st
( b3 = (b4 * decode ) | (code (Free c1)) & b4 in St c1,c2 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4 being Function of VAR ,c2 st
( b3 = (b4 * decode ) | (code (Free c1)) & b4 in St c1,c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Diagram ZF_FUND1:def 5 :
definition
let c
1 be
Universe;
let c
2 be
Subclass of c
1;
attr a
2 is
closed_wrt_A1 means :
Def6:
:: ZF_FUND1:def 6
for b
1 being
Element of a
1 holds
( b
1 in a
2 implies
{ {[(0-element_of a1),b2],[(1-element_of a1),b3]} where B is Element of a1, B is Element of a1 : ( b2 in b3 & b2 in b1 & b3 in b1 ) } in a
2 );
attr a
2 is
closed_wrt_A2 means :
Def7:
:: ZF_FUND1:def 7
for b
1, b
2 being
Element of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
{b1,b2} in a
2 );
attr a
2 is
closed_wrt_A3 means :
Def8:
:: ZF_FUND1:def 8
for b
1 being
Element of a
1 holds
( b
1 in a
2 implies
union b
1 in a
2 );
attr a
2 is
closed_wrt_A4 means :
Def9:
:: ZF_FUND1:def 9
for b
1, b
2 being
Element of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
{ {[b3,b4]} where B is Element of a1, B is Element of a1 : ( b3 in b1 & b4 in b2 ) } in a
2 );
attr a
2 is
closed_wrt_A5 means :
Def10:
:: ZF_FUND1:def 10
for b
1, b
2 being
Element of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
{ (b3 \/ b4) where B is Element of a1, B is Element of a1 : ( b3 in b1 & b4 in b2 ) } in a
2 );
attr a
2 is
closed_wrt_A6 means :
Def11:
:: ZF_FUND1:def 11
for b
1, b
2 being
Element of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
{ (b3 \ b4) where B is Element of a1, B is Element of a1 : ( b3 in b1 & b4 in b2 ) } in a
2 );
attr a
2 is
closed_wrt_A7 means :
Def12:
:: ZF_FUND1:def 12
for b
1, b
2 being
Element of a
1 holds
( b
1 in a
2 & b
2 in a
2 implies
{ (b3 (#) b4) where B is Element of a1, B is Element of a1 : ( b3 in b1 & b4 in b2 ) } in a
2 );
end;
:: deftheorem Def6 defines closed_wrt_A1 ZF_FUND1:def 6 :
:: deftheorem Def7 defines closed_wrt_A2 ZF_FUND1:def 7 :
:: deftheorem Def8 defines closed_wrt_A3 ZF_FUND1:def 8 :
:: deftheorem Def9 defines closed_wrt_A4 ZF_FUND1:def 9 :
:: deftheorem Def10 defines closed_wrt_A5 ZF_FUND1:def 10 :
:: deftheorem Def11 defines closed_wrt_A6 ZF_FUND1:def 11 :
:: deftheorem Def12 defines closed_wrt_A7 ZF_FUND1:def 12 :
:: deftheorem Def13 defines closed_wrt_A1-A7 ZF_FUND1:def 13 :
Lemma14:
for b1 being Element of omega holds b1 = x". (x. (card b1))
Lemma15:
for b1 being finite Subset of omega
for b2 being non empty set
for b3 being Function of VAR ,b2 holds
( dom ((b3 * decode ) | b1) = b1 & rng ((b3 * decode ) | b1) c= b2 & (b3 * decode ) | b1 in Funcs b1,b2 & dom (b3 * decode ) = omega )
Lemma16:
for b1 being non empty set
for b2 being Function of VAR ,b1
for b3 being Element of VAR holds
( decode . (x". b3) = b3 & (decode " ) . b3 = x". b3 & (b2 * decode ) . (x". b3) = b2 . b3 )
Lemma17:
for b1 being set
for b2 being finite Subset of VAR holds
( b1 in code b2 iff ex b3 being Element of VAR st
( b3 in b2 & b1 = x". b3 ) )
Lemma18:
for b1 being Element of VAR holds code {b1} = {(x". b1)}
Lemma19:
for b1, b2 being Element of VAR holds code {b1,b2} = {(x". b1),(x". b2)}
Lemma20:
for b1 being finite Subset of VAR holds b1, code b1 are_equipotent
Lemma21:
for b1, b2 being finite Subset of VAR holds
( code (b1 \/ b2) = (code b1) \/ (code b2) & code (b1 \ b2) = (code b1) \ (code b2) )
by RELAT_1:153, Lemma4, FUNCT_1:123;
Lemma22:
for b1 being non empty set
for b2 being Function of VAR ,b1
for b3 being Element of VAR
for b4 being ZF-formula holds
( b3 in Free b4 implies ((b2 * decode ) | (code (Free b4))) . (x". b3) = b2 . b3 )
Lemma23:
for b1 being non empty set
for b2 being ZF-formula
for b3, b4 being Function of VAR ,b1 holds
( (b3 * decode ) | (code (Free b2)) = (b4 * decode ) | (code (Free b2)) & b3 in St b2,b1 implies b4 in St b2,b1 )
Lemma24:
for b1 being set
for b2 being finite Subset of omega
for b3 being non empty set holds
not ( b1 in Funcs b2,b3 & ( for b4 being Function of VAR ,b3 holds
not b1 = (b4 * decode ) | b2 ) )
theorem Th1: :: ZF_FUND1:1
theorem Th2: :: ZF_FUND1:2
theorem Th3: :: ZF_FUND1:3
theorem Th4: :: ZF_FUND1:4
theorem Th5: :: ZF_FUND1:5
theorem Th6: :: ZF_FUND1:6
theorem Th7: :: ZF_FUND1:7
theorem Th8: :: ZF_FUND1:8
theorem Th9: :: ZF_FUND1:9
theorem Th10: :: ZF_FUND1:10
Lemma35:
for b1 being Universe
for b2 being Subclass of b1
for b3 being Element of omega holds
( b2 is closed_wrt_A1-A7 implies b3 in b2 )
Lemma36:
for b1 being Universe
for b2 being Subclass of b1 holds
( b2 is closed_wrt_A1-A7 implies ( 0-element_of b1 in b2 & 1-element_of b1 in b2 ) )
theorem Th11: :: ZF_FUND1:11
theorem Th12: :: ZF_FUND1:12
theorem Th13: :: ZF_FUND1:13
theorem Th14: :: ZF_FUND1:14
theorem Th15: :: ZF_FUND1:15
theorem Th16: :: ZF_FUND1:16
Lemma42:
for b1, b2, b3 being set holds {[b1,b2],[b2,b2]} (#) {[b2,b3]} = {[b1,b3],[b2,b3]}
theorem Th17: :: ZF_FUND1:17
Lemma44:
for b1, b2, b3, b4, b5, b6 being set holds
( b1 <> b2 implies {[b3,b1],[b4,b2]} (#) {[b1,b5],[b2,b6]} = {[b3,b5],[b4,b6]} )
Lemma45:
for b1, b2 being set
for b3 being Function holds
( dom b3 = {b1,b2} iff b3 = {[b1,(b3 . b1)],[b2,(b3 . b2)]} )
theorem Th18: :: ZF_FUND1:18
theorem Th19: :: ZF_FUND1:19
theorem Th20: :: ZF_FUND1:20
theorem Th21: :: ZF_FUND1:21
theorem Th22: :: ZF_FUND1:22
theorem Th23: :: ZF_FUND1:23
theorem Th24: :: ZF_FUND1:24
theorem Th25: :: ZF_FUND1:25
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 <> b
2 implies
{[b3,b1],[b4,b2]} (#) {[b1,b5],[b2,b6]} = {[b3,b5],[b4,b6]} )
by Lemma44;
theorem Th26: :: ZF_FUND1:26
canceled;
theorem Th27: :: ZF_FUND1:27
theorem Th28: :: ZF_FUND1:28
theorem Th29: :: ZF_FUND1:29
theorem Th30: :: ZF_FUND1:30
theorem Th31: :: ZF_FUND1:31
theorem Th32: :: ZF_FUND1:32
theorem Th33: :: ZF_FUND1:33
theorem Th34: :: ZF_FUND1:34
theorem Th35: :: ZF_FUND1:35
theorem Th36: :: ZF_FUND1:36
theorem Th37: :: ZF_FUND1:37
theorem Th38: :: ZF_FUND1:38