Step |
Hyp |
Ref |
Expression |
1 |
|
fvmptf.1 |
|- F/_ x A |
2 |
|
fvmptf.2 |
|- F/_ x C |
3 |
|
fvmptf.3 |
|- ( x = A -> B = C ) |
4 |
|
fvmptf.4 |
|- F = ( x e. D |-> B ) |
5 |
4
|
dmmptss |
|- dom F C_ D |
6 |
5
|
sseli |
|- ( A e. dom F -> A e. D ) |
7 |
|
eqid |
|- ( x e. D |-> ( _I ` B ) ) = ( x e. D |-> ( _I ` B ) ) |
8 |
4 7
|
fvmptex |
|- ( F ` A ) = ( ( x e. D |-> ( _I ` B ) ) ` A ) |
9 |
|
fvex |
|- ( _I ` C ) e. _V |
10 |
|
nfcv |
|- F/_ x _I |
11 |
10 2
|
nffv |
|- F/_ x ( _I ` C ) |
12 |
3
|
fveq2d |
|- ( x = A -> ( _I ` B ) = ( _I ` C ) ) |
13 |
1 11 12 7
|
fvmptf |
|- ( ( A e. D /\ ( _I ` C ) e. _V ) -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) ) |
14 |
9 13
|
mpan2 |
|- ( A e. D -> ( ( x e. D |-> ( _I ` B ) ) ` A ) = ( _I ` C ) ) |
15 |
8 14
|
eqtrid |
|- ( A e. D -> ( F ` A ) = ( _I ` C ) ) |
16 |
|
fvprc |
|- ( -. C e. _V -> ( _I ` C ) = (/) ) |
17 |
15 16
|
sylan9eq |
|- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) ) |
18 |
17
|
expcom |
|- ( -. C e. _V -> ( A e. D -> ( F ` A ) = (/) ) ) |
19 |
6 18
|
syl5 |
|- ( -. C e. _V -> ( A e. dom F -> ( F ` A ) = (/) ) ) |
20 |
|
ndmfv |
|- ( -. A e. dom F -> ( F ` A ) = (/) ) |
21 |
19 20
|
pm2.61d1 |
|- ( -. C e. _V -> ( F ` A ) = (/) ) |