Theorem freceq2 6664

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 109	. . . . . . . . 9 |- ( ( A = B /\ g e. _V ) -> A = B )
2	1	eleq2d 2308	. . . . . . . 8 |- ( ( A = B /\ g e. _V ) -> ( x e. A <-> x e. B ) )
3	2	anbi2d 468	. . . . . . 7 |- ( ( A = B /\ g e. _V ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom g = (/) /\ x e. B ) ) )
4	3	orbi2d 802	. . . . . 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 2358	. . . . 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 4221	. . . 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 6577	. . . 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 5062	. 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 6662	. 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 6662	. 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 2296	1 |- ( A = B -> frec ( F , A ) = frec ( F , B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821   (/)c0 3520   |-> cmpt 4192   suc csuc 4510   omcom 4737   dom cdm 4774   |` cres 4776   ` cfv 5377  recscrecs 6575  freccfrec 6661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-res 4786  df-iota 5337  df-fv 5385  df-recs 6576  df-frec 6662

This theorem is used by:  seqeq1  10887  seqeq3  10889