Theorem recseq 6085

Description: Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Assertion

Ref	Expression
recseq	|- ( F = G -> recs ( F ) = recs ( G ) )

Proof of Theorem recseq

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5317	. . . . . . . 8 |- ( F = G -> ( F ` ( a |` c ) ) = ( G ` ( a |` c ) ) )
2	1	eqeq2d 2100	. . . . . . 7 |- ( F = G -> ( ( a ` c ) = ( F ` ( a |` c ) ) <-> ( a ` c ) = ( G ` ( a |` c ) ) ) )
3	2	ralbidv 2381	. . . . . 6 |- ( F = G -> ( A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) <-> A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) )
4	3	anbi2d 453	. . . . 5 |- ( F = G -> ( ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) <-> ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) ) )
5	4	rexbidv 2382	. . . 4 |- ( F = G -> ( E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) <-> E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) ) )
6	5	abbidv 2206	. . 3 |- ( F = G -> { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) } = { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) } )
7	6	unieqd 3670	. 2 |- ( F = G -> U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) } = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) } )
8		df-recs 6084	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
9		df-recs 6084	. 2 |- recs ( G ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) }
10	7, 8, 9	3eqtr4g 2146	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   {cab 2075   A.wral 2360   E.wrex 2361   U.cuni 3659   Oncon0 4199   |` cres 4454   Fn wfn 5023   ` cfv 5028  recscrecs 6083

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-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-recs 6084

This theorem is referenced by:  rdgeq1  6150  rdgeq2  6151  freceq1  6171  freceq2  6172  frecsuclem  6185