Theorem rdgeq1 6537

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 5638	. . . . . 6 |- ( F = G -> ( F ` ( g ` x ) ) = ( G ` ( g ` x ) ) )
2	1	iuneq2d 3995	. . . . 5 |- ( F = G -> U_ x e. dom g ( F ` ( g ` x ) ) = U_ x e. dom g ( G ` ( g ` x ) ) )
3	2	uneq2d 3361	. . . 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 4180	. . 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 6472	. . 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 6536	. 2 |- rec ( F , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
8		df-irdg 6536	. 2 |- rec ( G , A ) = recs ( ( g e. _V |-> ( A u. U_ x e. dom g ( G ` ( g ` x ) ) ) ) )
9	6, 7, 8	3eqtr4g 2289	1 |- ( F = G -> rec ( F , A ) = rec ( G , A ) )

Syntax hints:   -> wi 4   = wceq 1397   _Vcvv 2802   u. cun 3198   U_ciun 3970   |-> cmpt 4150   dom cdm 4725   ` cfv 5326  recscrecs 6470   reccrdg 6535

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-iota 5286  df-fv 5334  df-recs 6471  df-irdg 6536

This theorem is referenced by:  omv  6623  oeiv  6624