Theorem recseq 6471

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 5638	. . . . . . . 8 |- ( F = G -> ( F ` ( a |` c ) ) = ( G ` ( a |` c ) ) )
2	1	eqeq2d 2243	. . . . . . 7 |- ( F = G -> ( ( a ` c ) = ( F ` ( a |` c ) ) <-> ( a ` c ) = ( G ` ( a |` c ) ) ) )
3	2	ralbidv 2532	. . . . . 6 |- ( F = G -> ( A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) <-> A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) )
4	3	anbi2d 464	. . . . 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 2533	. . . 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 2349	. . 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 3904	. 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 6470	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
9		df-recs 6470	. 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 2289	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   {cab 2217   A.wral 2510   E.wrex 2511   U.cuni 3893   Oncon0 4460   |` cres 4727   Fn wfn 5321   ` cfv 5326  recscrecs 6469

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-recs 6470

This theorem is referenced by:  rdgeq1  6536  rdgeq2  6537  freceq1  6557  freceq2  6558  frecsuclem  6571