Theorem freceq1 6450

Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)

Assertion

Ref	Expression
freceq1	|- ( F = G -> frec ( F , A ) = frec ( G , A ) )

Proof of Theorem freceq1

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

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . . . . . 11 |- ( ( F = G /\ g e. _V ) -> F = G )
2	1	fveq1d 5560	. . . . . . . . . 10 |- ( ( F = G /\ g e. _V ) -> ( F ` ( g ` m ) ) = ( G ` ( g ` m ) ) )
3	2	eleq2d 2266	. . . . . . . . 9 |- ( ( F = G /\ g e. _V ) -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( G ` ( g ` m ) ) ) )
4	3	anbi2d 464	. . . . . . . 8 |- ( ( F = G /\ g e. _V ) -> ( ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) ) )
5	4	rexbidv 2498	. . . . . . 7 |- ( ( F = G /\ g e. _V ) -> ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) ) )
6	5	orbi1d 792	. . . . . 6 |- ( ( F = G /\ g e. _V ) -> ( ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) ) )
7	6	abbidv 2314	. . . . 5 |- ( ( F = G /\ g e. _V ) -> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } = { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
8	7	mpteq2dva 4123	. . . 4 |- ( F = G -> ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )
9		recseq 6364	. . . 4 |- ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) -> recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) )
10	8, 9	syl 14	. . 3 |- ( F = G -> recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) )
11	10	reseq1d 4945	. 2 |- ( F = G -> (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om ) )
12		df-frec 6449	. 2 |- frec ( F , A ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om )
13		df-frec 6449	. 2 |- frec ( G , A ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( G ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om )
14	11, 12, 13	3eqtr4g 2254	1 |- ( F = G -> frec ( F , A ) = frec ( G , A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   {cab 2182   E.wrex 2476   _Vcvv 2763   (/)c0 3450   |-> cmpt 4094   suc csuc 4400   omcom 4626   dom cdm 4663   |` cres 4665   ` cfv 5258  recscrecs 6362  freccfrec 6448

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-res 4675  df-iota 5219  df-fv 5266  df-recs 6363  df-frec 6449

This theorem is referenced by:  frecuzrdgdom  10510  frecuzrdgfun  10512  frecuzrdgsuct  10516  seqeq1  10542  seqeq2  10543  seqeq3  10544  iseqvalcbv  10551  hashfz1  10875  ennnfonelemr  12640  ctinfom  12645  isomninn  15675  iswomninn  15694  ismkvnn  15697