Theorem recseq 6203

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 5420	. . . . . . . 8 |- ( F = G -> ( F ` ( a |` c ) ) = ( G ` ( a |` c ) ) )
2	1	eqeq2d 2151	. . . . . . 7 |- ( F = G -> ( ( a ` c ) = ( F ` ( a |` c ) ) <-> ( a ` c ) = ( G ` ( a |` c ) ) ) )
3	2	ralbidv 2437	. . . . . 6 |- ( F = G -> ( A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) <-> A. c e. b ( a ` c ) = ( G ` ( a |` c ) ) ) )
4	3	anbi2d 459	. . . . 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 2438	. . . 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 2257	. . 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 3747	. 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 6202	. 2 |- recs ( F ) = U. { a | E. b e. On ( a Fn b /\ A. c e. b ( a ` c ) = ( F ` ( a |` c ) ) ) }
9		df-recs 6202	. 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 2197	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   {cab 2125   A.wral 2416   E.wrex 2417   U.cuni 3736   Oncon0 4285   |` cres 4541   Fn wfn 5118   ` cfv 5123  recscrecs 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-recs 6202

This theorem is referenced by:  rdgeq1  6268  rdgeq2  6269  freceq1  6289  freceq2  6290  frecsuclem  6303