Step |
Hyp |
Ref |
Expression |
1 |
|
f1f |
|- ( F : A -1-1-> B -> F : A --> B ) |
2 |
|
fo2ndf |
|- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F ) |
3 |
1 2
|
syl |
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F -onto-> ran F ) |
4 |
|
f2ndf |
|- ( F : A --> B -> ( 2nd |` F ) : F --> B ) |
5 |
1 4
|
syl |
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F --> B ) |
6 |
|
fssxp |
|- ( F : A --> B -> F C_ ( A X. B ) ) |
7 |
1 6
|
syl |
|- ( F : A -1-1-> B -> F C_ ( A X. B ) ) |
8 |
|
ssel2 |
|- ( ( F C_ ( A X. B ) /\ x e. F ) -> x e. ( A X. B ) ) |
9 |
|
elxp2 |
|- ( x e. ( A X. B ) <-> E. a e. A E. v e. B x = <. a , v >. ) |
10 |
8 9
|
sylib |
|- ( ( F C_ ( A X. B ) /\ x e. F ) -> E. a e. A E. v e. B x = <. a , v >. ) |
11 |
|
ssel2 |
|- ( ( F C_ ( A X. B ) /\ y e. F ) -> y e. ( A X. B ) ) |
12 |
|
elxp2 |
|- ( y e. ( A X. B ) <-> E. b e. A E. w e. B y = <. b , w >. ) |
13 |
11 12
|
sylib |
|- ( ( F C_ ( A X. B ) /\ y e. F ) -> E. b e. A E. w e. B y = <. b , w >. ) |
14 |
10 13
|
anim12dan |
|- ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( E. a e. A E. v e. B x = <. a , v >. /\ E. b e. A E. w e. B y = <. b , w >. ) ) |
15 |
|
fvres |
|- ( <. a , v >. e. F -> ( ( 2nd |` F ) ` <. a , v >. ) = ( 2nd ` <. a , v >. ) ) |
16 |
15
|
ad2antrr |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( 2nd |` F ) ` <. a , v >. ) = ( 2nd ` <. a , v >. ) ) |
17 |
|
fvres |
|- ( <. b , w >. e. F -> ( ( 2nd |` F ) ` <. b , w >. ) = ( 2nd ` <. b , w >. ) ) |
18 |
17
|
ad2antlr |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( 2nd |` F ) ` <. b , w >. ) = ( 2nd ` <. b , w >. ) ) |
19 |
16 18
|
eqeq12d |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) <-> ( 2nd ` <. a , v >. ) = ( 2nd ` <. b , w >. ) ) ) |
20 |
|
vex |
|- a e. _V |
21 |
|
vex |
|- v e. _V |
22 |
20 21
|
op2nd |
|- ( 2nd ` <. a , v >. ) = v |
23 |
|
vex |
|- b e. _V |
24 |
|
vex |
|- w e. _V |
25 |
23 24
|
op2nd |
|- ( 2nd ` <. b , w >. ) = w |
26 |
22 25
|
eqeq12i |
|- ( ( 2nd ` <. a , v >. ) = ( 2nd ` <. b , w >. ) <-> v = w ) |
27 |
|
f1fun |
|- ( F : A -1-1-> B -> Fun F ) |
28 |
|
funopfv |
|- ( Fun F -> ( <. a , v >. e. F -> ( F ` a ) = v ) ) |
29 |
|
funopfv |
|- ( Fun F -> ( <. b , w >. e. F -> ( F ` b ) = w ) ) |
30 |
28 29
|
anim12d |
|- ( Fun F -> ( ( <. a , v >. e. F /\ <. b , w >. e. F ) -> ( ( F ` a ) = v /\ ( F ` b ) = w ) ) ) |
31 |
27 30
|
syl |
|- ( F : A -1-1-> B -> ( ( <. a , v >. e. F /\ <. b , w >. e. F ) -> ( ( F ` a ) = v /\ ( F ` b ) = w ) ) ) |
32 |
|
eqcom |
|- ( ( F ` a ) = v <-> v = ( F ` a ) ) |
33 |
32
|
biimpi |
|- ( ( F ` a ) = v -> v = ( F ` a ) ) |
34 |
|
eqcom |
|- ( ( F ` b ) = w <-> w = ( F ` b ) ) |
35 |
34
|
biimpi |
|- ( ( F ` b ) = w -> w = ( F ` b ) ) |
36 |
33 35
|
eqeqan12d |
|- ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( v = w <-> ( F ` a ) = ( F ` b ) ) ) |
37 |
|
simpl |
|- ( ( a e. A /\ v e. B ) -> a e. A ) |
38 |
|
simpl |
|- ( ( b e. A /\ w e. B ) -> b e. A ) |
39 |
37 38
|
anim12i |
|- ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( a e. A /\ b e. A ) ) |
40 |
|
f1veqaeq |
|- ( ( F : A -1-1-> B /\ ( a e. A /\ b e. A ) ) -> ( ( F ` a ) = ( F ` b ) -> a = b ) ) |
41 |
39 40
|
sylan2 |
|- ( ( F : A -1-1-> B /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( F ` a ) = ( F ` b ) -> a = b ) ) |
42 |
|
opeq12 |
|- ( ( a = b /\ v = w ) -> <. a , v >. = <. b , w >. ) |
43 |
42
|
ex |
|- ( a = b -> ( v = w -> <. a , v >. = <. b , w >. ) ) |
44 |
41 43
|
syl6 |
|- ( ( F : A -1-1-> B /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( F ` a ) = ( F ` b ) -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) |
45 |
44
|
com23 |
|- ( ( F : A -1-1-> B /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( v = w -> ( ( F ` a ) = ( F ` b ) -> <. a , v >. = <. b , w >. ) ) ) |
46 |
45
|
ex |
|- ( F : A -1-1-> B -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( v = w -> ( ( F ` a ) = ( F ` b ) -> <. a , v >. = <. b , w >. ) ) ) ) |
47 |
46
|
com14 |
|- ( ( F ` a ) = ( F ` b ) -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( v = w -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) ) |
48 |
36 47
|
syl6bi |
|- ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( v = w -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( v = w -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) ) ) |
49 |
48
|
com14 |
|- ( v = w -> ( v = w -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) ) ) |
50 |
49
|
pm2.43i |
|- ( v = w -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) ) |
51 |
50
|
com14 |
|- ( F : A -1-1-> B -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) ) |
52 |
51
|
com23 |
|- ( F : A -1-1-> B -> ( ( ( F ` a ) = v /\ ( F ` b ) = w ) -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) ) |
53 |
31 52
|
syld |
|- ( F : A -1-1-> B -> ( ( <. a , v >. e. F /\ <. b , w >. e. F ) -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) ) |
54 |
53
|
com13 |
|- ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( ( <. a , v >. e. F /\ <. b , w >. e. F ) -> ( F : A -1-1-> B -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) ) |
55 |
54
|
impcom |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( F : A -1-1-> B -> ( v = w -> <. a , v >. = <. b , w >. ) ) ) |
56 |
55
|
com23 |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( v = w -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) |
57 |
26 56
|
syl5bi |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( 2nd ` <. a , v >. ) = ( 2nd ` <. b , w >. ) -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) |
58 |
19 57
|
sylbid |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> ( F : A -1-1-> B -> <. a , v >. = <. b , w >. ) ) ) |
59 |
58
|
com23 |
|- ( ( ( <. a , v >. e. F /\ <. b , w >. e. F ) /\ ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) |
60 |
59
|
ex |
|- ( ( <. a , v >. e. F /\ <. b , w >. e. F ) -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) ) |
61 |
60
|
adantl |
|- ( ( F C_ ( A X. B ) /\ ( <. a , v >. e. F /\ <. b , w >. e. F ) ) -> ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) ) |
62 |
61
|
com12 |
|- ( ( ( a e. A /\ v e. B ) /\ ( b e. A /\ w e. B ) ) -> ( ( F C_ ( A X. B ) /\ ( <. a , v >. e. F /\ <. b , w >. e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) ) |
63 |
62
|
ad4ant13 |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( F C_ ( A X. B ) /\ ( <. a , v >. e. F /\ <. b , w >. e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) ) |
64 |
|
eleq1 |
|- ( x = <. a , v >. -> ( x e. F <-> <. a , v >. e. F ) ) |
65 |
64
|
ad2antlr |
|- ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) -> ( x e. F <-> <. a , v >. e. F ) ) |
66 |
|
eleq1 |
|- ( y = <. b , w >. -> ( y e. F <-> <. b , w >. e. F ) ) |
67 |
65 66
|
bi2anan9 |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( x e. F /\ y e. F ) <-> ( <. a , v >. e. F /\ <. b , w >. e. F ) ) ) |
68 |
67
|
anbi2d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) <-> ( F C_ ( A X. B ) /\ ( <. a , v >. e. F /\ <. b , w >. e. F ) ) ) ) |
69 |
|
fveq2 |
|- ( x = <. a , v >. -> ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` <. a , v >. ) ) |
70 |
69
|
ad2antlr |
|- ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) -> ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` <. a , v >. ) ) |
71 |
|
fveq2 |
|- ( y = <. b , w >. -> ( ( 2nd |` F ) ` y ) = ( ( 2nd |` F ) ` <. b , w >. ) ) |
72 |
70 71
|
eqeqan12d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) <-> ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) ) ) |
73 |
|
simpllr |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> x = <. a , v >. ) |
74 |
|
simpr |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> y = <. b , w >. ) |
75 |
73 74
|
eqeq12d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( x = y <-> <. a , v >. = <. b , w >. ) ) |
76 |
72 75
|
imbi12d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) <-> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) |
77 |
76
|
imbi2d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) <-> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` <. a , v >. ) = ( ( 2nd |` F ) ` <. b , w >. ) -> <. a , v >. = <. b , w >. ) ) ) ) |
78 |
63 68 77
|
3imtr4d |
|- ( ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) /\ y = <. b , w >. ) -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) |
79 |
78
|
ex |
|- ( ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) /\ ( b e. A /\ w e. B ) ) -> ( y = <. b , w >. -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) ) |
80 |
79
|
rexlimdvva |
|- ( ( ( a e. A /\ v e. B ) /\ x = <. a , v >. ) -> ( E. b e. A E. w e. B y = <. b , w >. -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) ) |
81 |
80
|
ex |
|- ( ( a e. A /\ v e. B ) -> ( x = <. a , v >. -> ( E. b e. A E. w e. B y = <. b , w >. -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) ) ) |
82 |
81
|
rexlimivv |
|- ( E. a e. A E. v e. B x = <. a , v >. -> ( E. b e. A E. w e. B y = <. b , w >. -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) ) |
83 |
82
|
imp |
|- ( ( E. a e. A E. v e. B x = <. a , v >. /\ E. b e. A E. w e. B y = <. b , w >. ) -> ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) |
84 |
14 83
|
mpcom |
|- ( ( F C_ ( A X. B ) /\ ( x e. F /\ y e. F ) ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) |
85 |
84
|
ex |
|- ( F C_ ( A X. B ) -> ( ( x e. F /\ y e. F ) -> ( F : A -1-1-> B -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) |
86 |
85
|
com23 |
|- ( F C_ ( A X. B ) -> ( F : A -1-1-> B -> ( ( x e. F /\ y e. F ) -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) ) |
87 |
7 86
|
mpcom |
|- ( F : A -1-1-> B -> ( ( x e. F /\ y e. F ) -> ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) |
88 |
87
|
ralrimivv |
|- ( F : A -1-1-> B -> A. x e. F A. y e. F ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) |
89 |
|
dff13 |
|- ( ( 2nd |` F ) : F -1-1-> B <-> ( ( 2nd |` F ) : F --> B /\ A. x e. F A. y e. F ( ( ( 2nd |` F ) ` x ) = ( ( 2nd |` F ) ` y ) -> x = y ) ) ) |
90 |
5 88 89
|
sylanbrc |
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-> B ) |
91 |
|
df-f1 |
|- ( ( 2nd |` F ) : F -1-1-> B <-> ( ( 2nd |` F ) : F --> B /\ Fun `' ( 2nd |` F ) ) ) |
92 |
91
|
simprbi |
|- ( ( 2nd |` F ) : F -1-1-> B -> Fun `' ( 2nd |` F ) ) |
93 |
90 92
|
syl |
|- ( F : A -1-1-> B -> Fun `' ( 2nd |` F ) ) |
94 |
|
dff1o3 |
|- ( ( 2nd |` F ) : F -1-1-onto-> ran F <-> ( ( 2nd |` F ) : F -onto-> ran F /\ Fun `' ( 2nd |` F ) ) ) |
95 |
3 93 94
|
sylanbrc |
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F ) |