Step |
Hyp |
Ref |
Expression |
1 |
|
rdgdmlim |
|- Lim dom rec ( F , A ) |
2 |
|
limomss |
|- ( Lim dom rec ( F , A ) -> _om C_ dom rec ( F , A ) ) |
3 |
1 2
|
ax-mp |
|- _om C_ dom rec ( F , A ) |
4 |
3
|
sseli |
|- ( B e. _om -> B e. dom rec ( F , A ) ) |
5 |
|
rdgsucg |
|- ( B e. dom rec ( F , A ) -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) ) |
6 |
4 5
|
syl |
|- ( B e. _om -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) ) |
7 |
|
peano2b |
|- ( B e. _om <-> suc B e. _om ) |
8 |
|
fvres |
|- ( suc B e. _om -> ( ( rec ( F , A ) |` _om ) ` suc B ) = ( rec ( F , A ) ` suc B ) ) |
9 |
7 8
|
sylbi |
|- ( B e. _om -> ( ( rec ( F , A ) |` _om ) ` suc B ) = ( rec ( F , A ) ` suc B ) ) |
10 |
|
fvres |
|- ( B e. _om -> ( ( rec ( F , A ) |` _om ) ` B ) = ( rec ( F , A ) ` B ) ) |
11 |
10
|
fveq2d |
|- ( B e. _om -> ( F ` ( ( rec ( F , A ) |` _om ) ` B ) ) = ( F ` ( rec ( F , A ) ` B ) ) ) |
12 |
6 9 11
|
3eqtr4d |
|- ( B e. _om -> ( ( rec ( F , A ) |` _om ) ` suc B ) = ( F ` ( ( rec ( F , A ) |` _om ) ` B ) ) ) |