Theorem rdgeq2 5959

Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
rdgeq2	|- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Proof of Theorem rdgeq2

Dummy variables x g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 3090	. . . 4 |- ( A = B -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) )
2	1	mpteq2dv 3848	. . 3 |- ( A = B -> ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) = ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
3		recseq 5921	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) = ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) -> recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) = recs ( ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) )
4	2, 3	syl 14	. 2 |- ( A = B -> recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) = recs ( ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) )
5		df-irdg 5957	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
6		df-irdg 5957	. 2 |- rec ( F , B ) = recs ( ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
7	4, 5, 6	3eqtr4g 2097	1 |- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Syntax hints:   -> wi 4   = wceq 1243   _Vcvv 2557   u. cun 2915   U_ciun 3657   |-> cmpt 3818   dom cdm 4345   ` cfv 4902  recscrecs 5919   reccrdg 5956

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-iota 4867  df-fv 4910  df-recs 5920  df-irdg 5957

This theorem is referenced by:  rdg0g  5975  oav  6034