Step |
Hyp |
Ref |
Expression |
1 |
|
cnveq |
|- ( x = (/) -> `' x = `' (/) ) |
2 |
1
|
eleq1d |
|- ( x = (/) -> ( `' x e. Fin <-> `' (/) e. Fin ) ) |
3 |
|
cnveq |
|- ( x = y -> `' x = `' y ) |
4 |
3
|
eleq1d |
|- ( x = y -> ( `' x e. Fin <-> `' y e. Fin ) ) |
5 |
|
cnveq |
|- ( x = ( y u. { z } ) -> `' x = `' ( y u. { z } ) ) |
6 |
5
|
eleq1d |
|- ( x = ( y u. { z } ) -> ( `' x e. Fin <-> `' ( y u. { z } ) e. Fin ) ) |
7 |
|
cnveq |
|- ( x = A -> `' x = `' A ) |
8 |
7
|
eleq1d |
|- ( x = A -> ( `' x e. Fin <-> `' A e. Fin ) ) |
9 |
|
cnv0 |
|- `' (/) = (/) |
10 |
|
0fin |
|- (/) e. Fin |
11 |
9 10
|
eqeltri |
|- `' (/) e. Fin |
12 |
|
cnvun |
|- `' ( y u. { z } ) = ( `' y u. `' { z } ) |
13 |
|
elvv |
|- ( z e. ( _V X. _V ) <-> E. u E. v z = <. u , v >. ) |
14 |
|
sneq |
|- ( z = <. u , v >. -> { z } = { <. u , v >. } ) |
15 |
|
cnveq |
|- ( { z } = { <. u , v >. } -> `' { z } = `' { <. u , v >. } ) |
16 |
|
vex |
|- u e. _V |
17 |
|
vex |
|- v e. _V |
18 |
16 17
|
cnvsn |
|- `' { <. u , v >. } = { <. v , u >. } |
19 |
15 18
|
eqtrdi |
|- ( { z } = { <. u , v >. } -> `' { z } = { <. v , u >. } ) |
20 |
14 19
|
syl |
|- ( z = <. u , v >. -> `' { z } = { <. v , u >. } ) |
21 |
|
snfi |
|- { <. v , u >. } e. Fin |
22 |
20 21
|
eqeltrdi |
|- ( z = <. u , v >. -> `' { z } e. Fin ) |
23 |
22
|
exlimivv |
|- ( E. u E. v z = <. u , v >. -> `' { z } e. Fin ) |
24 |
13 23
|
sylbi |
|- ( z e. ( _V X. _V ) -> `' { z } e. Fin ) |
25 |
|
dfdm4 |
|- dom { z } = ran `' { z } |
26 |
|
dmsnn0 |
|- ( z e. ( _V X. _V ) <-> dom { z } =/= (/) ) |
27 |
26
|
biimpri |
|- ( dom { z } =/= (/) -> z e. ( _V X. _V ) ) |
28 |
27
|
necon1bi |
|- ( -. z e. ( _V X. _V ) -> dom { z } = (/) ) |
29 |
25 28
|
eqtr3id |
|- ( -. z e. ( _V X. _V ) -> ran `' { z } = (/) ) |
30 |
|
relcnv |
|- Rel `' { z } |
31 |
|
relrn0 |
|- ( Rel `' { z } -> ( `' { z } = (/) <-> ran `' { z } = (/) ) ) |
32 |
30 31
|
ax-mp |
|- ( `' { z } = (/) <-> ran `' { z } = (/) ) |
33 |
29 32
|
sylibr |
|- ( -. z e. ( _V X. _V ) -> `' { z } = (/) ) |
34 |
33 10
|
eqeltrdi |
|- ( -. z e. ( _V X. _V ) -> `' { z } e. Fin ) |
35 |
24 34
|
pm2.61i |
|- `' { z } e. Fin |
36 |
|
unfi |
|- ( ( `' y e. Fin /\ `' { z } e. Fin ) -> ( `' y u. `' { z } ) e. Fin ) |
37 |
35 36
|
mpan2 |
|- ( `' y e. Fin -> ( `' y u. `' { z } ) e. Fin ) |
38 |
12 37
|
eqeltrid |
|- ( `' y e. Fin -> `' ( y u. { z } ) e. Fin ) |
39 |
38
|
a1i |
|- ( y e. Fin -> ( `' y e. Fin -> `' ( y u. { z } ) e. Fin ) ) |
40 |
2 4 6 8 11 39
|
findcard2 |
|- ( A e. Fin -> `' A e. Fin ) |