Theorem rdgeq2 6199

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 3170	. . . 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 3959	. . 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 6133	. . 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 6197	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
6		df-irdg 6197	. 2 |- rec ( F , B ) = recs ( ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
7	4, 5, 6	3eqtr4g 2157	1 |- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Syntax hints:   -> wi 4   = wceq 1299   _Vcvv 2641   u. cun 3019   U_ciun 3760   |-> cmpt 3929   dom cdm 4477   ` cfv 5059  recscrecs 6131   reccrdg 6196

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-iota 5024  df-fv 5067  df-recs 6132  df-irdg 6197

This theorem is referenced by:  rdg0g  6215  oav  6280