Step |
Hyp |
Ref |
Expression |
1 |
|
fvsnun.1 |
|- ( ph -> A e. V ) |
2 |
|
fvsnun.2 |
|- ( ph -> B e. W ) |
3 |
|
fvsnun.3 |
|- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |
4 |
3
|
reseq1i |
|- ( G |` { A } ) = ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` { A } ) |
5 |
|
resundir |
|- ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` { A } ) = ( ( { <. A , B >. } |` { A } ) u. ( ( F |` ( C \ { A } ) ) |` { A } ) ) |
6 |
|
incom |
|- ( ( C \ { A } ) i^i { A } ) = ( { A } i^i ( C \ { A } ) ) |
7 |
|
disjdif |
|- ( { A } i^i ( C \ { A } ) ) = (/) |
8 |
6 7
|
eqtri |
|- ( ( C \ { A } ) i^i { A } ) = (/) |
9 |
|
resdisj |
|- ( ( ( C \ { A } ) i^i { A } ) = (/) -> ( ( F |` ( C \ { A } ) ) |` { A } ) = (/) ) |
10 |
8 9
|
ax-mp |
|- ( ( F |` ( C \ { A } ) ) |` { A } ) = (/) |
11 |
10
|
uneq2i |
|- ( ( { <. A , B >. } |` { A } ) u. ( ( F |` ( C \ { A } ) ) |` { A } ) ) = ( ( { <. A , B >. } |` { A } ) u. (/) ) |
12 |
|
un0 |
|- ( ( { <. A , B >. } |` { A } ) u. (/) ) = ( { <. A , B >. } |` { A } ) |
13 |
11 12
|
eqtri |
|- ( ( { <. A , B >. } |` { A } ) u. ( ( F |` ( C \ { A } ) ) |` { A } ) ) = ( { <. A , B >. } |` { A } ) |
14 |
5 13
|
eqtri |
|- ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` { A } ) = ( { <. A , B >. } |` { A } ) |
15 |
4 14
|
eqtri |
|- ( G |` { A } ) = ( { <. A , B >. } |` { A } ) |
16 |
15
|
fveq1i |
|- ( ( G |` { A } ) ` A ) = ( ( { <. A , B >. } |` { A } ) ` A ) |
17 |
|
snidg |
|- ( A e. V -> A e. { A } ) |
18 |
1 17
|
syl |
|- ( ph -> A e. { A } ) |
19 |
18
|
fvresd |
|- ( ph -> ( ( G |` { A } ) ` A ) = ( G ` A ) ) |
20 |
18
|
fvresd |
|- ( ph -> ( ( { <. A , B >. } |` { A } ) ` A ) = ( { <. A , B >. } ` A ) ) |
21 |
|
fvsng |
|- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B ) |
22 |
1 2 21
|
syl2anc |
|- ( ph -> ( { <. A , B >. } ` A ) = B ) |
23 |
20 22
|
eqtrd |
|- ( ph -> ( ( { <. A , B >. } |` { A } ) ` A ) = B ) |
24 |
16 19 23
|
3eqtr3a |
|- ( ph -> ( G ` A ) = B ) |