Theorem rdgeq1 6615

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 5674	. . . . . 6 |- ( F = G -> ( F ` ( g ` x ) ) = ( G ` ( g ` x ) ) )
2	1	iuneq2d 4021	. . . . 5 |- ( F = G -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom g ( G ` ( g ` x ) ) )
3	2	uneq2d 3377	. . . 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 4206	. . 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 6550	. . 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 6614	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
8		df-irdg 6614	. 2 |- rec ( G , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
9	6, 7, 8	3eqtr4g 2292	1 |- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 2815   u. cun 3212   U_ciun 3996   |-> cmpt 4176   dom cdm 4754   ` cfv 5357  recscrecs 6548   reccrdg 6613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-iota 5317  df-fv 5365  df-recs 6549  df-irdg 6614

This theorem is referenced by:  omv  6701  oeiv  6702