Theorem rdgeq1 6276

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 5428	. . . . . 6 |- ( F = G -> ( F ` ( g ` x ) ) = ( G ` ( g ` x ) ) )
2	1	iuneq2d 3846	. . . . 5 |- ( F = G -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom g ( G ` ( g ` x ) ) )
3	2	uneq2d 3235	. . . 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 4027	. . 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 6211	. . 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 6275	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
8		df-irdg 6275	. 2 |- rec ( G , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
9	6, 7, 8	3eqtr4g 2198	1 |- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Syntax hints:   -> wi 4   = wceq 1332   _Vcvv 2689   u. cun 3074   U_ciun 3821   |-> cmpt 3997   dom cdm 4547   ` cfv 5131  recscrecs 6209   reccrdg 6274

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-iota 5096  df-fv 5139  df-recs 6210  df-irdg 6275

This theorem is referenced by:  omv  6359  oeiv  6360