Theorem freceq2 6172

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

Assertion

Ref	Expression
freceq2	|- ( A = B -> frec ( F , A ) = frec ( F , B ) )

Proof of Theorem freceq2

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

Step	Hyp	Ref	Expression
1		simpl 108	. . . . . . . . 9 |- ( ( A = B /\ g e. _V ) -> A = B )
2	1	eleq2d 2158	. . . . . . . 8 |- ( ( A = B /\ g e. _V ) -> ( x e. A <-> x e. B ) )
3	2	anbi2d 453	. . . . . . 7 |- ( ( A = B /\ g e. _V ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom g = (/) /\ x e. B ) ) )
4	3	orbi2d 740	. . . . . 6 |- ( ( A = B /\ 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) ) )
5	4	abbidv 2206	. . . . 5 |- ( ( A = B /\ 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } )
6	5	mpteq2dva 3934	. . . 4 |- ( A = B -> ( 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) )
7		recseq 6085	. . . 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) -> 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) ) )
8	6, 7	syl 14	. . 3 |- ( A = B -> 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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) ) )
9	8	reseq1d 4725	. 2 |- ( A = B -> (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. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) ) |` om ) )
10		df-frec 6170	. 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 )
11		df-frec 6170	. 2 |- frec ( F , B ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. B ) ) } ) ) |` om )
12	9, 10, 11	3eqtr4g 2146	1 |- ( A = B -> frec ( F , A ) = frec ( F , B ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 665   = wceq 1290   e. wcel 1439   {cab 2075   E.wrex 2361   _Vcvv 2620   (/)c0 3287   |-> cmpt 3905   suc csuc 4201   omcom 4418   dom cdm 4452   |` cres 4454   ` cfv 5028  recscrecs 6083  freccfrec 6169

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-res 4464  df-iota 4993  df-fv 5036  df-recs 6084  df-frec 6170

This theorem is referenced by:  iseqeq1  9919  iseqeq3  9921