Theorem freceq1 5979

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 102	. . . . . . . . . . 11 |- ( ( F = G /\ g e. _V ) -> F = G )
2	1	fveq1d 5180	. . . . . . . . . 10 |- ( ( F = G /\ g e. _V ) -> ( F ` ( g ` m ) ) = ( G ` ( g ` m ) ) )
3	2	eleq2d 2107	. . . . . . . . 9 |- ( ( F = G /\ g e. _V ) -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( G ` ( g ` m ) ) ) )
4	3	anbi2d 437	. . . . . . . 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 2327	. . . . . . 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 705	. . . . . 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 2155	. . . . 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 3847	. . . 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 5921	. . . 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 4611	. 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 5978	. 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 5978	. 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 2097	1 |- ( F = G -> frec ( F , A ) = frec ( G , A ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   (/)c0 3224   |-> cmpt 3818   suc csuc 4102   omcom 4313   dom cdm 4345   |` cres 4347   ` cfv 4902  recscrecs 5919  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-res 4357  df-iota 4867  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  iseqeq1  9214  iseqeq2  9215  iseqeq3  9216  iseqeq4  9217  iseqval  9220