Step |
Hyp |
Ref |
Expression |
1 |
|
f1ocnv |
|- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A ) |
2 |
|
f1of |
|- ( `' F : B -1-1-onto-> A -> `' F : B --> A ) |
3 |
1 2
|
syl |
|- ( F : A -1-1-onto-> B -> `' F : B --> A ) |
4 |
|
feu |
|- ( ( `' F : B --> A /\ C e. B ) -> E! x e. A <. C , x >. e. `' F ) |
5 |
3 4
|
sylan |
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A <. C , x >. e. `' F ) |
6 |
|
f1ocnvfvb |
|- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) ) |
7 |
6
|
3com23 |
|- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) ) |
8 |
|
dff1o4 |
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) ) |
9 |
8
|
simprbi |
|- ( F : A -1-1-onto-> B -> `' F Fn B ) |
10 |
|
fnopfvb |
|- ( ( `' F Fn B /\ C e. B ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) ) |
11 |
10
|
3adant3 |
|- ( ( `' F Fn B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) ) |
12 |
9 11
|
syl3an1 |
|- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) ) |
13 |
7 12
|
bitrd |
|- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) ) |
14 |
13
|
3expa |
|- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) ) |
15 |
14
|
reubidva |
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( E! x e. A ( F ` x ) = C <-> E! x e. A <. C , x >. e. `' F ) ) |
16 |
5 15
|
mpbird |
|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C ) |