| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dnnumch.f |
|- F = recs ( ( z e. _V |-> ( G ` ( A \ ran z ) ) ) ) |
| 2 |
|
dnnumch.a |
|- ( ph -> A e. V ) |
| 3 |
|
dnnumch.g |
|- ( ph -> A. y e. ~P A ( y =/= (/) -> ( G ` y ) e. y ) ) |
| 4 |
|
eqid |
|- ( x e. A |-> |^| ( `' F " { x } ) ) = ( x e. A |-> |^| ( `' F " { x } ) ) |
| 5 |
|
sneq |
|- ( x = w -> { x } = { w } ) |
| 6 |
5
|
imaeq2d |
|- ( x = w -> ( `' F " { x } ) = ( `' F " { w } ) ) |
| 7 |
6
|
inteqd |
|- ( x = w -> |^| ( `' F " { x } ) = |^| ( `' F " { w } ) ) |
| 8 |
|
simpr |
|- ( ( ph /\ w e. A ) -> w e. A ) |
| 9 |
|
cnvimass |
|- ( `' F " { w } ) C_ dom F |
| 10 |
1
|
tfr1 |
|- F Fn On |
| 11 |
10
|
fndmi |
|- dom F = On |
| 12 |
9 11
|
sseqtri |
|- ( `' F " { w } ) C_ On |
| 13 |
1 2 3
|
dnnumch2 |
|- ( ph -> A C_ ran F ) |
| 14 |
13
|
sselda |
|- ( ( ph /\ w e. A ) -> w e. ran F ) |
| 15 |
|
inisegn0 |
|- ( w e. ran F <-> ( `' F " { w } ) =/= (/) ) |
| 16 |
14 15
|
sylib |
|- ( ( ph /\ w e. A ) -> ( `' F " { w } ) =/= (/) ) |
| 17 |
|
oninton |
|- ( ( ( `' F " { w } ) C_ On /\ ( `' F " { w } ) =/= (/) ) -> |^| ( `' F " { w } ) e. On ) |
| 18 |
12 16 17
|
sylancr |
|- ( ( ph /\ w e. A ) -> |^| ( `' F " { w } ) e. On ) |
| 19 |
4 7 8 18
|
fvmptd3 |
|- ( ( ph /\ w e. A ) -> ( ( x e. A |-> |^| ( `' F " { x } ) ) ` w ) = |^| ( `' F " { w } ) ) |