:: FUNCT_4 semantic presentation
Lemma1:
for b1, b2, b3 being set holds
( [b1,b2] in b3 implies ( b1 in union (union b3) & b2 in union (union b3) ) )
Lemma2:
for b1, b2, b3, b4, b5, b6, b7, b8 being set holds
( [[b1,b2],[b3,b4]] = [[b5,b6],[b7,b8]] implies ( b1 = b5 & b3 = b7 & b2 = b6 & b4 = b8 ) )
theorem Th1: :: FUNCT_4:1
for b
1 being
set holds
not ( ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4 being
set holds
not b
2 = [b3,b4] ) ) ) & ( for b
2, b
3 being
set holds
not b
1 c= [:b2,b3:] ) )
theorem Th2: :: FUNCT_4:2
theorem Th3: :: FUNCT_4:3
canceled;
theorem Th4: :: FUNCT_4:4
for b
1, b
2 being
set holds
(
id b
1 c= id b
2 iff b
1 c= b
2 )
theorem Th5: :: FUNCT_4:5
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 --> b
3 c= b
2 --> b
3 )
theorem Th6: :: FUNCT_4:6
for b
1, b
2, b
3, b
4 being
set holds
( b
1 --> b
2 c= b
3 --> b
4 implies b
1 c= b
3 )
theorem Th7: :: FUNCT_4:7
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
1 --> b
2 c= b
3 --> b
4 implies b
2 = b
4 )
theorem Th8: :: FUNCT_4:8
theorem Th9: :: FUNCT_4:9
theorem Th10: :: FUNCT_4:10
for b
1, b
2 being
set for b
3, b
4 being
Function holds
( b
3 <= b
4 implies
(b1 | b3) | b
2 <= (b1 | b4) | b
2 )
:: deftheorem Def1 defines +* FUNCT_4:def 1 :
theorem Th11: :: FUNCT_4:11
theorem Th12: :: FUNCT_4:12
theorem Th13: :: FUNCT_4:13
theorem Th14: :: FUNCT_4:14
theorem Th15: :: FUNCT_4:15
theorem Th16: :: FUNCT_4:16
theorem Th17: :: FUNCT_4:17
theorem Th18: :: FUNCT_4:18
theorem Th19: :: FUNCT_4:19
theorem Th20: :: FUNCT_4:20
theorem Th21: :: FUNCT_4:21
theorem Th22: :: FUNCT_4:22
theorem Th23: :: FUNCT_4:23
theorem Th24: :: FUNCT_4:24
theorem Th25: :: FUNCT_4:25
theorem Th26: :: FUNCT_4:26
theorem Th27: :: FUNCT_4:27
theorem Th28: :: FUNCT_4:28
theorem Th29: :: FUNCT_4:29
theorem Th30: :: FUNCT_4:30
theorem Th31: :: FUNCT_4:31
theorem Th32: :: FUNCT_4:32
theorem Th33: :: FUNCT_4:33
theorem Th34: :: FUNCT_4:34
theorem Th35: :: FUNCT_4:35
theorem Th36: :: FUNCT_4:36
theorem Th37: :: FUNCT_4:37
for b
1, b
2 being
set for b
3, b
4 being
PartFunc of b
1,b
2 holds
( b
4 is
total implies b
3 +* b
4 = b
4 )
theorem Th38: :: FUNCT_4:38
for b
1, b
2 being
set for b
3, b
4 being
Function of b
1,b
2 holds
( ( b
2 = {} implies b
1 = {} ) implies b
3 +* b
4 = b
4 )
theorem Th39: :: FUNCT_4:39
for b
1 being
set for b
2, b
3 being
Function of b
1,b
1 holds b
2 +* b
3 = b
3
theorem Th40: :: FUNCT_4:40
theorem Th41: :: FUNCT_4:41
definition
let c
1 be
Function;
func ~ c
1 -> Function means :
Def2:
:: FUNCT_4:def 2
( ( for b
1 being
set holds
( b
1 in dom a
2 iff ex b
2, b
3 being
set st
( b
1 = [b3,b2] &
[b2,b3] in dom a
1 ) ) ) & ( for b
1, b
2 being
set holds
(
[b1,b2] in dom a
1 implies a
2 . [b2,b1] = a
1 . [b1,b2] ) ) );
existence
ex b1 being Function st
( ( for b2 being set holds
( b2 in dom b1 iff ex b3, b4 being set st
( b2 = [b4,b3] & [b3,b4] in dom c1 ) ) ) & ( for b2, b3 being set holds
( [b2,b3] in dom c1 implies b1 . [b3,b2] = c1 . [b2,b3] ) ) )
uniqueness
for b1, b2 being Function holds
( ( for b3 being set holds
( b3 in dom b1 iff ex b4, b5 being set st
( b3 = [b5,b4] & [b4,b5] in dom c1 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in dom c1 implies b1 . [b4,b3] = c1 . [b3,b4] ) ) & ( for b3 being set holds
( b3 in dom b2 iff ex b4, b5 being set st
( b3 = [b5,b4] & [b4,b5] in dom c1 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in dom c1 implies b2 . [b4,b3] = c1 . [b3,b4] ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines ~ FUNCT_4:def 2 :
for b
1, b
2 being
Function holds
( b
2 = ~ b
1 iff ( ( for b
3 being
set holds
( b
3 in dom b
2 iff ex b
4, b
5 being
set st
( b
3 = [b5,b4] &
[b4,b5] in dom b
1 ) ) ) & ( for b
3, b
4 being
set holds
(
[b3,b4] in dom b
1 implies b
2 . [b4,b3] = b
1 . [b3,b4] ) ) ) );
theorem Th42: :: FUNCT_4:42
theorem Th43: :: FUNCT_4:43
theorem Th44: :: FUNCT_4:44
theorem Th45: :: FUNCT_4:45
theorem Th46: :: FUNCT_4:46
theorem Th47: :: FUNCT_4:47
theorem Th48: :: FUNCT_4:48
theorem Th49: :: FUNCT_4:49
theorem Th50: :: FUNCT_4:50
theorem Th51: :: FUNCT_4:51
theorem Th52: :: FUNCT_4:52
theorem Th53: :: FUNCT_4:53
theorem Th54: :: FUNCT_4:54
theorem Th55: :: FUNCT_4:55
canceled;
theorem Th56: :: FUNCT_4:56
canceled;
definition
let c
1, c
2 be
Function;
func |:c1,c2:| -> Function means :
Def3:
:: FUNCT_4:def 3
( ( for b
1 being
set holds
( b
1 in dom a
3 iff ex b
2, b
3, b
4, b
5 being
set st
( b
1 = [[b2,b4],[b3,b5]] &
[b2,b3] in dom a
1 &
[b4,b5] in dom a
2 ) ) ) & ( for b
1, b
2, b
3, b
4 being
set holds
(
[b1,b2] in dom a
1 &
[b3,b4] in dom a
2 implies a
3 . [[b1,b3],[b2,b4]] = [(a1 . [b1,b2]),(a2 . [b3,b4])] ) ) );
existence
ex b1 being Function st
( ( for b2 being set holds
( b2 in dom b1 iff ex b3, b4, b5, b6 being set st
( b2 = [[b3,b5],[b4,b6]] & [b3,b4] in dom c1 & [b5,b6] in dom c2 ) ) ) & ( for b2, b3, b4, b5 being set holds
( [b2,b3] in dom c1 & [b4,b5] in dom c2 implies b1 . [[b2,b4],[b3,b5]] = [(c1 . [b2,b3]),(c2 . [b4,b5])] ) ) )
uniqueness
for b1, b2 being Function holds
( ( for b3 being set holds
( b3 in dom b1 iff ex b4, b5, b6, b7 being set st
( b3 = [[b4,b6],[b5,b7]] & [b4,b5] in dom c1 & [b6,b7] in dom c2 ) ) ) & ( for b3, b4, b5, b6 being set holds
( [b3,b4] in dom c1 & [b5,b6] in dom c2 implies b1 . [[b3,b5],[b4,b6]] = [(c1 . [b3,b4]),(c2 . [b5,b6])] ) ) & ( for b3 being set holds
( b3 in dom b2 iff ex b4, b5, b6, b7 being set st
( b3 = [[b4,b6],[b5,b7]] & [b4,b5] in dom c1 & [b6,b7] in dom c2 ) ) ) & ( for b3, b4, b5, b6 being set holds
( [b3,b4] in dom c1 & [b5,b6] in dom c2 implies b2 . [[b3,b5],[b4,b6]] = [(c1 . [b3,b4]),(c2 . [b5,b6])] ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines |: FUNCT_4:def 3 :
for b
1, b
2, b
3 being
Function holds
( b
3 = |:b1,b2:| iff ( ( for b
4 being
set holds
( b
4 in dom b
3 iff ex b
5, b
6, b
7, b
8 being
set st
( b
4 = [[b5,b7],[b6,b8]] &
[b5,b6] in dom b
1 &
[b7,b8] in dom b
2 ) ) ) & ( for b
4, b
5, b
6, b
7 being
set holds
(
[b4,b5] in dom b
1 &
[b6,b7] in dom b
2 implies b
3 . [[b4,b6],[b5,b7]] = [(b1 . [b4,b5]),(b2 . [b6,b7])] ) ) ) );
theorem Th57: :: FUNCT_4:57
for b
1, b
2, b
3, b
4 being
set for b
5, b
6 being
Function holds
(
[[b1,b2],[b3,b4]] in dom |:b5,b6:| iff (
[b1,b3] in dom b
5 &
[b2,b4] in dom b
6 ) )
theorem Th58: :: FUNCT_4:58
for b
1, b
2, b
3, b
4 being
set for b
5, b
6 being
Function holds
(
[[b1,b2],[b3,b4]] in dom |:b5,b6:| implies
|:b5,b6:| . [[b1,b2],[b3,b4]] = [(b5 . [b1,b3]),(b6 . [b2,b4])] )
theorem Th59: :: FUNCT_4:59
theorem Th60: :: FUNCT_4:60
for b
1, b
2, b
3, b
4 being
set for b
5, b
6 being
Function holds
(
dom b
5 c= [:b1,b2:] &
dom b
6 c= [:b3,b4:] implies
dom |:b5,b6:| c= [:[:b1,b3:],[:b2,b4:]:] )
theorem Th61: :: FUNCT_4:61
for b
1, b
2, b
3, b
4 being
set for b
5, b
6 being
Function holds
(
dom b
5 = [:b1,b2:] &
dom b
6 = [:b3,b4:] implies
dom |:b5,b6:| = [:[:b1,b3:],[:b2,b4:]:] )
theorem Th62: :: FUNCT_4:62
for b
1, b
2, b
3, b
4, b
5, b
6 being
set for b
7 being
PartFunc of
[:b1,b2:],b
3for b
8 being
PartFunc of
[:b4,b5:],b
6 holds
|:b7,b8:| is
PartFunc of
[:[:b1,b4:],[:b2,b5:]:],
[:b3,b6:]
theorem Th63: :: FUNCT_4:63
for b
1, b
2, b
3, b
4, b
5, b
6 being
set for b
7 being
Function of
[:b1,b2:],b
3for b
8 being
Function of
[:b4,b5:],b
6 holds
( b
3 <> {} & b
6 <> {} implies
|:b7,b8:| is
Function of
[:[:b1,b4:],[:b2,b5:]:],
[:b3,b6:] )
theorem Th64: :: FUNCT_4:64
for b
1, b
2, b
3, b
4 being
set for b
5, b
6 being non
empty set for b
7 being
Function of
[:b1,b2:],b
5for b
8 being
Function of
[:b3,b4:],b
6 holds
|:b7,b8:| is
Function of
[:[:b1,b3:],[:b2,b4:]:],
[:b5,b6:] by Th63;
:: deftheorem Def4 defines --> FUNCT_4:def 4 :
theorem Th65: :: FUNCT_4:65
theorem Th66: :: FUNCT_4:66
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> b
2 implies (
(b1,b2 --> b3,b4) . b
1 = b
3 &
(b1,b2 --> b3,b4) . b
2 = b
4 ) )
theorem Th67: :: FUNCT_4:67
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> b
2 implies
rng (b1,b2 --> b3,b4) = {b3,b4} )
theorem Th68: :: FUNCT_4:68
for b
1, b
2, b
3 being
set holds b
1,b
2 --> b
3,b
3 = {b1,b2} --> b
3
definition
let c
1 be non
empty set ;
let c
2, c
3 be
set ;
let c
4, c
5 be
Element of c
1;
redefine func --> as c
2,c
3 --> c
4,c
5 -> Function of
{a2,a3},a
1;
coherence
c2,c3 --> c4,c5 is Function of {c2,c3},c1
end;
theorem Th69: :: FUNCT_4:69
for b
1, b
2, b
3, b
4 being
set for b
5 being
Function holds
(
dom b
5 = {b1,b2} & b
5 . b
1 = b
3 & b
5 . b
2 = b
4 implies b
5 = b
1,b
2 --> b
3,b
4 )
theorem Th70: :: FUNCT_4:70
theorem Th71: :: FUNCT_4:71
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> b
3 implies b
1,b
3 --> b
2,b
4 = {[b1,b2],[b3,b4]} )
theorem Th72: :: FUNCT_4:72
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 <> b
2 & b
1,b
2 --> b
3,b
4 = b
1,b
2 --> b
5,b
6 implies ( b
3 = b
5 & b
4 = b
6 ) )
theorem Th73: :: FUNCT_4:73
theorem Th74: :: FUNCT_4:74
theorem Th75: :: FUNCT_4:75
theorem Th76: :: FUNCT_4:76
theorem Th77: :: FUNCT_4:77
theorem Th78: :: FUNCT_4:78
theorem Th79: :: FUNCT_4:79
for b
1, b
2 being
Function holds
( b
1 c= b
2 implies ( b
1 +* b
2 = b
2 & b
2 +* b
1 = b
2 ) )
theorem Th80: :: FUNCT_4:80
theorem Th81: :: FUNCT_4:81
theorem Th82: :: FUNCT_4:82
theorem Th83: :: FUNCT_4:83