Theorem freceq2 6372

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 2240	. . . . . . . 8 |- ( ( A = B /\ g e. _V ) -> ( x e. A <-> x e. B ) )
3	2	anbi2d 461	. . . . . . 7 |- ( ( A = B /\ g e. _V ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom g = (/) /\ x e. B ) ) )
4	3	orbi2d 785	. . . . . 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 2288	. . . . 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 4079	. . . 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 6285	. . . 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 4890	. 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 6370	. 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 6370	. 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 2228	1 |- ( A = B -> frec ( F , A ) = frec ( F , B ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730   (/)c0 3414   |-> cmpt 4050   suc csuc 4350   omcom 4574   dom cdm 4611   |` cres 4613   ` cfv 5198  recscrecs 6283  freccfrec 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-res 4623  df-iota 5160  df-fv 5206  df-recs 6284  df-frec 6370

This theorem is referenced by:  seqeq1  10404  seqeq3  10406