| Step |
Hyp |
Ref |
Expression |
| 1 |
|
initopropdlemlem.1 |
|- F Fn X |
| 2 |
|
initopropdlemlem.2 |
|- ( ph -> -. A e. Y ) |
| 3 |
|
initopropdlemlem.3 |
|- X C_ Y |
| 4 |
|
initopropdlemlem.4 |
|- ( ( ph /\ B e. X ) -> ( F ` B ) = (/) ) |
| 5 |
3
|
sseli |
|- ( A e. X -> A e. Y ) |
| 6 |
2 5
|
nsyl |
|- ( ph -> -. A e. X ) |
| 7 |
1
|
fndmi |
|- dom F = X |
| 8 |
7
|
eleq2i |
|- ( A e. dom F <-> A e. X ) |
| 9 |
|
ndmfv |
|- ( -. A e. dom F -> ( F ` A ) = (/) ) |
| 10 |
8 9
|
sylnbir |
|- ( -. A e. X -> ( F ` A ) = (/) ) |
| 11 |
6 10
|
syl |
|- ( ph -> ( F ` A ) = (/) ) |
| 12 |
11
|
adantr |
|- ( ( ph /\ B e. X ) -> ( F ` A ) = (/) ) |
| 13 |
12 4
|
eqtr4d |
|- ( ( ph /\ B e. X ) -> ( F ` A ) = ( F ` B ) ) |
| 14 |
11
|
adantr |
|- ( ( ph /\ -. B e. X ) -> ( F ` A ) = (/) ) |
| 15 |
7
|
eleq2i |
|- ( B e. dom F <-> B e. X ) |
| 16 |
|
ndmfv |
|- ( -. B e. dom F -> ( F ` B ) = (/) ) |
| 17 |
15 16
|
sylnbir |
|- ( -. B e. X -> ( F ` B ) = (/) ) |
| 18 |
17
|
adantl |
|- ( ( ph /\ -. B e. X ) -> ( F ` B ) = (/) ) |
| 19 |
14 18
|
eqtr4d |
|- ( ( ph /\ -. B e. X ) -> ( F ` A ) = ( F ` B ) ) |
| 20 |
13 19
|
pm2.61dan |
|- ( ph -> ( F ` A ) = ( F ` B ) ) |