Step |
Hyp |
Ref |
Expression |
1 |
|
rdgsucmptf.1 |
|- F/_ x A |
2 |
|
rdgsucmptf.2 |
|- F/_ x B |
3 |
|
rdgsucmptf.3 |
|- F/_ x D |
4 |
|
rdgsucmptf.4 |
|- F = rec ( ( x e. _V |-> C ) , A ) |
5 |
|
rdgsucmptf.5 |
|- ( x = ( F ` B ) -> C = D ) |
6 |
4
|
fveq1i |
|- ( F ` suc B ) = ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) |
7 |
|
rdgdmlim |
|- Lim dom rec ( ( x e. _V |-> C ) , A ) |
8 |
|
limsuc |
|- ( Lim dom rec ( ( x e. _V |-> C ) , A ) -> ( B e. dom rec ( ( x e. _V |-> C ) , A ) <-> suc B e. dom rec ( ( x e. _V |-> C ) , A ) ) ) |
9 |
7 8
|
ax-mp |
|- ( B e. dom rec ( ( x e. _V |-> C ) , A ) <-> suc B e. dom rec ( ( x e. _V |-> C ) , A ) ) |
10 |
|
rdgsucg |
|- ( B e. dom rec ( ( x e. _V |-> C ) , A ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) ) ) |
11 |
4
|
fveq1i |
|- ( F ` B ) = ( rec ( ( x e. _V |-> C ) , A ) ` B ) |
12 |
11
|
fveq2i |
|- ( ( x e. _V |-> C ) ` ( F ` B ) ) = ( ( x e. _V |-> C ) ` ( rec ( ( x e. _V |-> C ) , A ) ` B ) ) |
13 |
10 12
|
eqtr4di |
|- ( B e. dom rec ( ( x e. _V |-> C ) , A ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = ( ( x e. _V |-> C ) ` ( F ` B ) ) ) |
14 |
|
nfmpt1 |
|- F/_ x ( x e. _V |-> C ) |
15 |
14 1
|
nfrdg |
|- F/_ x rec ( ( x e. _V |-> C ) , A ) |
16 |
4 15
|
nfcxfr |
|- F/_ x F |
17 |
16 2
|
nffv |
|- F/_ x ( F ` B ) |
18 |
|
eqid |
|- ( x e. _V |-> C ) = ( x e. _V |-> C ) |
19 |
17 3 5 18
|
fvmptnf |
|- ( -. D e. _V -> ( ( x e. _V |-> C ) ` ( F ` B ) ) = (/) ) |
20 |
13 19
|
sylan9eqr |
|- ( ( -. D e. _V /\ B e. dom rec ( ( x e. _V |-> C ) , A ) ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = (/) ) |
21 |
20
|
ex |
|- ( -. D e. _V -> ( B e. dom rec ( ( x e. _V |-> C ) , A ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = (/) ) ) |
22 |
9 21
|
syl5bir |
|- ( -. D e. _V -> ( suc B e. dom rec ( ( x e. _V |-> C ) , A ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = (/) ) ) |
23 |
|
ndmfv |
|- ( -. suc B e. dom rec ( ( x e. _V |-> C ) , A ) -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = (/) ) |
24 |
22 23
|
pm2.61d1 |
|- ( -. D e. _V -> ( rec ( ( x e. _V |-> C ) , A ) ` suc B ) = (/) ) |
25 |
6 24
|
eqtrid |
|- ( -. D e. _V -> ( F ` suc B ) = (/) ) |