Step |
Hyp |
Ref |
Expression |
1 |
|
hashgval.1 |
|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) |
2 |
|
resundir |
|- ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) ) |` Fin ) = ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |` Fin ) u. ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) ) |
3 |
|
eqid |
|- ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) |
4 |
|
eqid |
|- ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
5 |
3 4
|
hashkf |
|- ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) : Fin --> NN0 |
6 |
|
ffn |
|- ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) : Fin --> NN0 -> ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) Fn Fin ) |
7 |
|
fnresdm |
|- ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) Fn Fin -> ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |` Fin ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) ) |
8 |
5 6 7
|
mp2b |
|- ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |` Fin ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
9 |
|
disjdifr |
|- ( ( _V \ Fin ) i^i Fin ) = (/) |
10 |
|
pnfex |
|- +oo e. _V |
11 |
10
|
fconst |
|- ( ( _V \ Fin ) X. { +oo } ) : ( _V \ Fin ) --> { +oo } |
12 |
|
ffn |
|- ( ( ( _V \ Fin ) X. { +oo } ) : ( _V \ Fin ) --> { +oo } -> ( ( _V \ Fin ) X. { +oo } ) Fn ( _V \ Fin ) ) |
13 |
|
fnresdisj |
|- ( ( ( _V \ Fin ) X. { +oo } ) Fn ( _V \ Fin ) -> ( ( ( _V \ Fin ) i^i Fin ) = (/) <-> ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/) ) ) |
14 |
11 12 13
|
mp2b |
|- ( ( ( _V \ Fin ) i^i Fin ) = (/) <-> ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/) ) |
15 |
9 14
|
mpbi |
|- ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) = (/) |
16 |
8 15
|
uneq12i |
|- ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |` Fin ) u. ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) ) = ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. (/) ) |
17 |
|
un0 |
|- ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. (/) ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
18 |
16 17
|
eqtri |
|- ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |` Fin ) u. ( ( ( _V \ Fin ) X. { +oo } ) |` Fin ) ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
19 |
2 18
|
eqtri |
|- ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) ) |` Fin ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
20 |
|
df-hash |
|- # = ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) ) |
21 |
20
|
reseq1i |
|- ( # |` Fin ) = ( ( ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) u. ( ( _V \ Fin ) X. { +oo } ) ) |` Fin ) |
22 |
1
|
coeq1i |
|- ( G o. card ) = ( ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` _om ) o. card ) |
23 |
19 21 22
|
3eqtr4i |
|- ( # |` Fin ) = ( G o. card ) |
24 |
23
|
fveq1i |
|- ( ( # |` Fin ) ` A ) = ( ( G o. card ) ` A ) |
25 |
|
cardf2 |
|- card : { x | E. y e. On y ~~ x } --> On |
26 |
|
ffun |
|- ( card : { x | E. y e. On y ~~ x } --> On -> Fun card ) |
27 |
25 26
|
ax-mp |
|- Fun card |
28 |
|
finnum |
|- ( A e. Fin -> A e. dom card ) |
29 |
|
fvco |
|- ( ( Fun card /\ A e. dom card ) -> ( ( G o. card ) ` A ) = ( G ` ( card ` A ) ) ) |
30 |
27 28 29
|
sylancr |
|- ( A e. Fin -> ( ( G o. card ) ` A ) = ( G ` ( card ` A ) ) ) |
31 |
24 30
|
eqtrid |
|- ( A e. Fin -> ( ( # |` Fin ) ` A ) = ( G ` ( card ` A ) ) ) |
32 |
|
fvres |
|- ( A e. Fin -> ( ( # |` Fin ) ` A ) = ( # ` A ) ) |
33 |
31 32
|
eqtr3d |
|- ( A e. Fin -> ( G ` ( card ` A ) ) = ( # ` A ) ) |