Theorem freceq2 6394

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 2247	. . . . . . . 8 |- ( ( A = B /\ g e. _V ) -> ( x e. A <-> x e. B ) )
3	2	anbi2d 464	. . . . . . 7 |- ( ( A = B /\ g e. _V ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom g = (/) /\ x e. B ) ) )
4	3	orbi2d 790	. . . . . 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 2295	. . . . 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 4094	. . . 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 6307	. . . 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 4907	. 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 6392	. 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 6392	. 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 2235	1 |- ( A = B -> frec ( F , A ) = frec ( F , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2738   (/)c0 3423   |-> cmpt 4065   suc csuc 4366   omcom 4590   dom cdm 4627   |` cres 4629   ` cfv 5217  recscrecs 6305  freccfrec 6391

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-in 3136  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-res 4639  df-iota 5179  df-fv 5225  df-recs 6306  df-frec 6392

This theorem is referenced by:  seqeq1  10448  seqeq3  10450