Theorem rdgeq1 5981

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

Assertion

Ref	Expression
rdgeq1	|- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Proof of Theorem rdgeq1

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

Step	Hyp	Ref	Expression
1		fveq1 5197	. . . . . 6 |- ( F = G -> ( F ` ( g ` x ) ) = ( G ` ( g ` x ) ) )
2	1	iuneq2d 3703	. . . . 5 |- ( F = G -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom g ( G ` ( g ` x ) ) )
3	2	uneq2d 3126	. . . 4 |- ( F = G -> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) = ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) )
4	3	mpteq2dv 3869	. . 3 |- ( F = G -> ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) = ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
5		recseq 5944	. . 3 |- ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) = ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) -> recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) ) )
6	4, 5	syl 14	. 2 |- ( F = G -> recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) ) )
7		df-irdg 5980	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
8		df-irdg 5980	. 2 |- rec ( G , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
9	6, 7, 8	3eqtr4g 2138	1 |- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Syntax hints:   -> wi 4   = wceq 1284   _Vcvv 2601   u. cun 2971   U_ciun 3678   |-> cmpt 3839   dom cdm 4363   ` cfv 4922  recscrecs 5942   reccrdg 5979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-iota 4887  df-fv 4930  df-recs 5943  df-irdg 5980

This theorem is referenced by:  omv  6058  oeiv  6059