Step |
Hyp |
Ref |
Expression |
1 |
|
abrexexg |
|- ( A e. V -> { z | E. x e. A z = ( card ` B ) } e. _V ) |
2 |
|
vex |
|- y e. _V |
3 |
|
eqeq1 |
|- ( z = y -> ( z = ( card ` B ) <-> y = ( card ` B ) ) ) |
4 |
3
|
rexbidv |
|- ( z = y -> ( E. x e. A z = ( card ` B ) <-> E. x e. A y = ( card ` B ) ) ) |
5 |
2 4
|
elab |
|- ( y e. { z | E. x e. A z = ( card ` B ) } <-> E. x e. A y = ( card ` B ) ) |
6 |
|
cardidm |
|- ( card ` ( card ` B ) ) = ( card ` B ) |
7 |
|
fveq2 |
|- ( y = ( card ` B ) -> ( card ` y ) = ( card ` ( card ` B ) ) ) |
8 |
|
id |
|- ( y = ( card ` B ) -> y = ( card ` B ) ) |
9 |
6 7 8
|
3eqtr4a |
|- ( y = ( card ` B ) -> ( card ` y ) = y ) |
10 |
9
|
rexlimivw |
|- ( E. x e. A y = ( card ` B ) -> ( card ` y ) = y ) |
11 |
5 10
|
sylbi |
|- ( y e. { z | E. x e. A z = ( card ` B ) } -> ( card ` y ) = y ) |
12 |
11
|
rgen |
|- A. y e. { z | E. x e. A z = ( card ` B ) } ( card ` y ) = y |
13 |
|
carduni |
|- ( { z | E. x e. A z = ( card ` B ) } e. _V -> ( A. y e. { z | E. x e. A z = ( card ` B ) } ( card ` y ) = y -> ( card ` U. { z | E. x e. A z = ( card ` B ) } ) = U. { z | E. x e. A z = ( card ` B ) } ) ) |
14 |
1 12 13
|
mpisyl |
|- ( A e. V -> ( card ` U. { z | E. x e. A z = ( card ` B ) } ) = U. { z | E. x e. A z = ( card ` B ) } ) |
15 |
|
fvex |
|- ( card ` B ) e. _V |
16 |
15
|
dfiun2 |
|- U_ x e. A ( card ` B ) = U. { z | E. x e. A z = ( card ` B ) } |
17 |
16
|
fveq2i |
|- ( card ` U_ x e. A ( card ` B ) ) = ( card ` U. { z | E. x e. A z = ( card ` B ) } ) |
18 |
14 17 16
|
3eqtr4g |
|- ( A e. V -> ( card ` U_ x e. A ( card ` B ) ) = U_ x e. A ( card ` B ) ) |
19 |
18
|
adantr |
|- ( ( A e. V /\ A. x e. A ( card ` B ) = B ) -> ( card ` U_ x e. A ( card ` B ) ) = U_ x e. A ( card ` B ) ) |
20 |
|
iuneq2 |
|- ( A. x e. A ( card ` B ) = B -> U_ x e. A ( card ` B ) = U_ x e. A B ) |
21 |
20
|
adantl |
|- ( ( A e. V /\ A. x e. A ( card ` B ) = B ) -> U_ x e. A ( card ` B ) = U_ x e. A B ) |
22 |
21
|
fveq2d |
|- ( ( A e. V /\ A. x e. A ( card ` B ) = B ) -> ( card ` U_ x e. A ( card ` B ) ) = ( card ` U_ x e. A B ) ) |
23 |
19 22 21
|
3eqtr3d |
|- ( ( A e. V /\ A. x e. A ( card ` B ) = B ) -> ( card ` U_ x e. A B ) = U_ x e. A B ) |
24 |
23
|
ex |
|- ( A e. V -> ( A. x e. A ( card ` B ) = B -> ( card ` U_ x e. A B ) = U_ x e. A B ) ) |