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Theorem recseq 6476
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.
StepHypRef Expression
1 fveq1 5607 . . . . . . . 8  |-  ( F  =  G  ->  ( F `  ( a  |`  c ) )  =  ( G `  (
a  |`  c ) ) )
21eqeq2d 2369 . . . . . . 7  |-  ( F  =  G  ->  (
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
32ralbidv 2639 . . . . . 6  |-  ( F  =  G  ->  ( A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
43anbi2d 684 . . . . 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 ) ) ) ) )
54rexbidv 2640 . . . 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 ) ) ) ) )
65abbidv 2472 . . 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 ) ) ) } )
76unieqd 3919 . 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 6475 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
9 df-recs 6475 . 2  |- recs ( G )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
107, 8, 93eqtr4g 2415 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642   {cab 2344   A.wral 2619   E.wrex 2620   U.cuni 3908   Oncon0 4474    |` cres 4773    Fn wfn 5332   ` cfv 5337  recscrecs 6474
This theorem is referenced by:  rdgeq1  6511  rdgeq2  6512  dfoi  7316  oieq1  7317  oieq2  7318  ordtypecbv  7322  dfac12r  7862  zorn2g  8220  ttukey2g  8233  aomclem3  26476  aomclem8  26482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-recs 6475
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