Theorem recseq 6321

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 5526	. . . . . . . 8 |- ( F = G -> ( F ` ( a |` c ) ) = ( G ` ( a |` c ) ) )
2	1	eqeq2d 2199	. . . . . . 7 |- ( F = G -> ( ( a ` c ) = ( F ` ( a |` c ) ) <-> ( a ` c ) = ( G ` ( a |` c ) ) ) )
3	2	ralbidv 2487	. . . . . 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 2488	. . . 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 2305	. . 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 3832	. 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 6320	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
9		df-recs 6320	. 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 2245	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   {cab 2173   A.wral 2465   E.wrex 2466   U.cuni 3821   Oncon0 4375   |` cres 4640   Fn wfn 5223   ` cfv 5228  recscrecs 6319

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-recs 6320

This theorem is referenced by:  rdgeq1  6386  rdgeq2  6387  freceq1  6407  freceq2  6408  frecsuclem  6421