Step |
Hyp |
Ref |
Expression |
1 |
|
funimaeq.x |
|- F/ x ph |
2 |
|
funimaeq.f |
|- ( ph -> Fun F ) |
3 |
|
funimaeq.g |
|- ( ph -> Fun G ) |
4 |
|
funimaeq.a |
|- ( ph -> A C_ dom F ) |
5 |
|
funimaeq.d |
|- ( ph -> A C_ dom G ) |
6 |
|
funimaeq.e |
|- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) ) |
7 |
3
|
funfnd |
|- ( ph -> G Fn dom G ) |
8 |
7
|
adantr |
|- ( ( ph /\ x e. A ) -> G Fn dom G ) |
9 |
5
|
adantr |
|- ( ( ph /\ x e. A ) -> A C_ dom G ) |
10 |
|
simpr |
|- ( ( ph /\ x e. A ) -> x e. A ) |
11 |
|
fnfvima |
|- ( ( G Fn dom G /\ A C_ dom G /\ x e. A ) -> ( G ` x ) e. ( G " A ) ) |
12 |
8 9 10 11
|
syl3anc |
|- ( ( ph /\ x e. A ) -> ( G ` x ) e. ( G " A ) ) |
13 |
6 12
|
eqeltrd |
|- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( G " A ) ) |
14 |
1 2 13
|
funimassd |
|- ( ph -> ( F " A ) C_ ( G " A ) ) |
15 |
2
|
funfnd |
|- ( ph -> F Fn dom F ) |
16 |
15
|
adantr |
|- ( ( ph /\ x e. A ) -> F Fn dom F ) |
17 |
4
|
adantr |
|- ( ( ph /\ x e. A ) -> A C_ dom F ) |
18 |
|
fnfvima |
|- ( ( F Fn dom F /\ A C_ dom F /\ x e. A ) -> ( F ` x ) e. ( F " A ) ) |
19 |
16 17 10 18
|
syl3anc |
|- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( F " A ) ) |
20 |
6 19
|
eqeltrrd |
|- ( ( ph /\ x e. A ) -> ( G ` x ) e. ( F " A ) ) |
21 |
1 3 20
|
funimassd |
|- ( ph -> ( G " A ) C_ ( F " A ) ) |
22 |
14 21
|
eqssd |
|- ( ph -> ( F " A ) = ( G " A ) ) |