Theorem rdgeq1 6339

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 5485	. . . . . 6 |- ( F = G -> ( F ` ( g ` x ) ) = ( G ` ( g ` x ) ) )
2	1	iuneq2d 3891	. . . . 5 |- ( F = G -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom g ( G ` ( g ` x ) ) )
3	2	uneq2d 3276	. . . 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 4073	. . 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 6274	. . 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 6338	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
8		df-irdg 6338	. 2 |- rec ( G , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
9	6, 7, 8	3eqtr4g 2224	1 |- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   u. cun 3114   U_ciun 3866   |-> cmpt 4043   dom cdm 4604   ` cfv 5188  recscrecs 6272   reccrdg 6337

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-iota 5153  df-fv 5196  df-recs 6273  df-irdg 6338

This theorem is referenced by:  omv  6423  oeiv  6424