Step |
Hyp |
Ref |
Expression |
1 |
|
fsneq.a |
|- ( ph -> A e. V ) |
2 |
|
fsneq.b |
|- B = { A } |
3 |
|
fsneq.f |
|- ( ph -> F Fn B ) |
4 |
|
fsneq.g |
|- ( ph -> G Fn B ) |
5 |
|
eqfnfv |
|- ( ( F Fn B /\ G Fn B ) -> ( F = G <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) |
6 |
3 4 5
|
syl2anc |
|- ( ph -> ( F = G <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) |
7 |
|
snidg |
|- ( A e. V -> A e. { A } ) |
8 |
1 7
|
syl |
|- ( ph -> A e. { A } ) |
9 |
2
|
eqcomi |
|- { A } = B |
10 |
9
|
a1i |
|- ( ph -> { A } = B ) |
11 |
8 10
|
eleqtrd |
|- ( ph -> A e. B ) |
12 |
11
|
adantr |
|- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> A e. B ) |
13 |
|
simpr |
|- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> A. x e. B ( F ` x ) = ( G ` x ) ) |
14 |
|
fveq2 |
|- ( x = A -> ( F ` x ) = ( F ` A ) ) |
15 |
|
fveq2 |
|- ( x = A -> ( G ` x ) = ( G ` A ) ) |
16 |
14 15
|
eqeq12d |
|- ( x = A -> ( ( F ` x ) = ( G ` x ) <-> ( F ` A ) = ( G ` A ) ) ) |
17 |
16
|
rspcva |
|- ( ( A e. B /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> ( F ` A ) = ( G ` A ) ) |
18 |
12 13 17
|
syl2anc |
|- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> ( F ` A ) = ( G ` A ) ) |
19 |
18
|
ex |
|- ( ph -> ( A. x e. B ( F ` x ) = ( G ` x ) -> ( F ` A ) = ( G ` A ) ) ) |
20 |
|
simpl |
|- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` A ) = ( G ` A ) ) |
21 |
2
|
eleq2i |
|- ( x e. B <-> x e. { A } ) |
22 |
21
|
biimpi |
|- ( x e. B -> x e. { A } ) |
23 |
|
velsn |
|- ( x e. { A } <-> x = A ) |
24 |
22 23
|
sylib |
|- ( x e. B -> x = A ) |
25 |
24
|
fveq2d |
|- ( x e. B -> ( F ` x ) = ( F ` A ) ) |
26 |
25
|
adantl |
|- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` x ) = ( F ` A ) ) |
27 |
24
|
fveq2d |
|- ( x e. B -> ( G ` x ) = ( G ` A ) ) |
28 |
27
|
adantl |
|- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( G ` x ) = ( G ` A ) ) |
29 |
20 26 28
|
3eqtr4d |
|- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) ) |
30 |
29
|
adantll |
|- ( ( ( ph /\ ( F ` A ) = ( G ` A ) ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) ) |
31 |
30
|
ralrimiva |
|- ( ( ph /\ ( F ` A ) = ( G ` A ) ) -> A. x e. B ( F ` x ) = ( G ` x ) ) |
32 |
31
|
ex |
|- ( ph -> ( ( F ` A ) = ( G ` A ) -> A. x e. B ( F ` x ) = ( G ` x ) ) ) |
33 |
19 32
|
impbid |
|- ( ph -> ( A. x e. B ( F ` x ) = ( G ` x ) <-> ( F ` A ) = ( G ` A ) ) ) |
34 |
6 33
|
bitrd |
|- ( ph -> ( F = G <-> ( F ` A ) = ( G ` A ) ) ) |