Theorem freceq1 6480

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 5580	. . . . . . . . . 10 |- ( ( F = G /\ g e. _V ) -> ( F ` ( g ` m ) ) = ( G ` ( g ` m ) ) )
3	2	eleq2d 2275	. . . . . . . . 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 2507	. . . . . . 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 793	. . . . . 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 2323	. . . . 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 4135	. . . 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 6394	. . . 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 4959	. 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 6479	. 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 6479	. 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 2263	1 |- ( F = G -> frec ( F , A ) = frec ( G , A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   (/)c0 3460   |-> cmpt 4106   suc csuc 4413   omcom 4639   dom cdm 4676   |` cres 4678   ` cfv 5272  recscrecs 6392  freccfrec 6478

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-res 4688  df-iota 5233  df-fv 5280  df-recs 6393  df-frec 6479

This theorem is referenced by:  frecuzrdgdom  10565  frecuzrdgfun  10567  frecuzrdgsuct  10571  seqeq1  10597  seqeq2  10598  seqeq3  10599  iseqvalcbv  10606  hashfz1  10930  ennnfonelemr  12827  ctinfom  12832  isomninn  16007  iswomninn  16026  ismkvnn  16029