Theorem rdgeq2 6319

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 3254	. . . 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 4055	. . 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 6253	. . 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 6317	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
6		df-irdg 6317	. 2 |- rec ( F , B ) = recs ( ( g e. _V |-> ( B u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
7	4, 5, 6	3eqtr4g 2215	1 |- ( A = B -> rec ( F , A ) = rec ( F , B ) )

Syntax hints:   -> wi 4   = wceq 1335   _Vcvv 2712   u. cun 3100   U_ciun 3849   |-> cmpt 4025   dom cdm 4586   ` cfv 5170  recscrecs 6251   reccrdg 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-iota 5135  df-fv 5178  df-recs 6252  df-irdg 6317

This theorem is referenced by:  rdg0g  6335  oav  6401