Theorem freceq1 6538

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 5629	. . . . . . . . . 10 |- ( ( F = G /\ g e. _V ) -> ( F ` ( g ` m ) ) = ( G ` ( g ` m ) ) )
3	2	eleq2d 2299	. . . . . . . . 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 2531	. . . . . . 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 796	. . . . . 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 2347	. . . . 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 4174	. . . 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 6452	. . . 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 5004	. 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 6537	. 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 6537	. 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 2287	1 |- ( F = G -> frec ( F , A ) = frec ( G , A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   (/)c0 3491   |-> cmpt 4145   suc csuc 4456   omcom 4682   dom cdm 4719   |` cres 4721   ` cfv 5318  recscrecs 6450  freccfrec 6536

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-res 4731  df-iota 5278  df-fv 5326  df-recs 6451  df-frec 6537

This theorem is referenced by:  frecuzrdgdom  10640  frecuzrdgfun  10642  frecuzrdgsuct  10646  seqeq1  10672  seqeq2  10673  seqeq3  10674  iseqvalcbv  10681  hashfz1  11005  ennnfonelemr  12994  ctinfom  12999  isomninn  16399  iswomninn  16418  ismkvnn  16421