Step |
Hyp |
Ref |
Expression |
1 |
|
fvmptg.1 |
|- ( x = A -> B = C ) |
2 |
|
fvmptg.2 |
|- F = ( x e. D |-> B ) |
3 |
1 2
|
fvmptg |
|- ( ( A e. D /\ C e. _V ) -> ( F ` A ) = C ) |
4 |
|
fvi |
|- ( C e. _V -> ( _I ` C ) = C ) |
5 |
4
|
adantl |
|- ( ( A e. D /\ C e. _V ) -> ( _I ` C ) = C ) |
6 |
3 5
|
eqtr4d |
|- ( ( A e. D /\ C e. _V ) -> ( F ` A ) = ( _I ` C ) ) |
7 |
1
|
eleq1d |
|- ( x = A -> ( B e. _V <-> C e. _V ) ) |
8 |
2
|
dmmpt |
|- dom F = { x e. D | B e. _V } |
9 |
7 8
|
elrab2 |
|- ( A e. dom F <-> ( A e. D /\ C e. _V ) ) |
10 |
9
|
baib |
|- ( A e. D -> ( A e. dom F <-> C e. _V ) ) |
11 |
10
|
notbid |
|- ( A e. D -> ( -. A e. dom F <-> -. C e. _V ) ) |
12 |
|
ndmfv |
|- ( -. A e. dom F -> ( F ` A ) = (/) ) |
13 |
11 12
|
syl6bir |
|- ( A e. D -> ( -. C e. _V -> ( F ` A ) = (/) ) ) |
14 |
13
|
imp |
|- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) ) |
15 |
|
fvprc |
|- ( -. C e. _V -> ( _I ` C ) = (/) ) |
16 |
15
|
adantl |
|- ( ( A e. D /\ -. C e. _V ) -> ( _I ` C ) = (/) ) |
17 |
14 16
|
eqtr4d |
|- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = ( _I ` C ) ) |
18 |
6 17
|
pm2.61dan |
|- ( A e. D -> ( F ` A ) = ( _I ` C ) ) |