Theorem recseq 6550

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 5674	. . . . . . . 8 |- ( F = G -> ( F ` ( a |` c ) ) = ( G ` ( a |` c ) ) )
2	1	eqeq2d 2246	. . . . . . 7 |- ( F = G -> ( ( a ` c ) = ( F ` ( a |` c ) ) <-> ( a ` c ) = ( G ` ( a |` c ) ) ) )
3	2	ralbidv 2544	. . . . . 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 2545	. . . 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 2354	. . 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 3930	. 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 6549	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
9		df-recs 6549	. 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 2292	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   {cab 2220   A.wral 2522   E.wrex 2523   U.cuni 3919   Oncon0 4489   |` cres 4756   Fn wfn 5352   ` cfv 5357  recscrecs 6548

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-recs 6549

This theorem is referenced by:  rdgeq1  6615  rdgeq2  6616  freceq1  6636  freceq2  6637  frecsuclem  6650