Step |
Hyp |
Ref |
Expression |
1 |
|
funres |
|- ( Fun A -> Fun ( A |` B ) ) |
2 |
1
|
3ad2ant1 |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> Fun ( A |` B ) ) |
3 |
|
finresfin |
|- ( A e. Fin -> ( A |` B ) e. Fin ) |
4 |
3
|
3ad2ant2 |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( A |` B ) e. Fin ) |
5 |
|
hashfun |
|- ( ( A |` B ) e. Fin -> ( Fun ( A |` B ) <-> ( # ` ( A |` B ) ) = ( # ` dom ( A |` B ) ) ) ) |
6 |
4 5
|
syl |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( Fun ( A |` B ) <-> ( # ` ( A |` B ) ) = ( # ` dom ( A |` B ) ) ) ) |
7 |
2 6
|
mpbid |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` ( A |` B ) ) = ( # ` dom ( A |` B ) ) ) |
8 |
|
ssdmres |
|- ( B C_ dom A <-> dom ( A |` B ) = B ) |
9 |
8
|
biimpi |
|- ( B C_ dom A -> dom ( A |` B ) = B ) |
10 |
9
|
3ad2ant3 |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> dom ( A |` B ) = B ) |
11 |
10
|
fveq2d |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` dom ( A |` B ) ) = ( # ` B ) ) |
12 |
7 11
|
eqtrd |
|- ( ( Fun A /\ A e. Fin /\ B C_ dom A ) -> ( # ` ( A |` B ) ) = ( # ` B ) ) |