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Theorem recseq 7055
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 5871 . . . . . . . 8  |-  ( F  =  G  ->  ( F `  ( a  |`  c ) )  =  ( G `  (
a  |`  c ) ) )
21eqeq2d 2481 . . . . . . 7  |-  ( F  =  G  ->  (
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
32ralbidv 2906 . . . . . 6  |-  ( F  =  G  ->  ( A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
43anbi2d 703 . . . . 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 2978 . . . 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 2603 . . 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 4261 . 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 7054 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
9 df-recs 7054 . 2  |- recs ( G )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
107, 8, 93eqtr4g 2533 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   {cab 2452   A.wral 2817   E.wrex 2818   U.cuni 4251   Oncon0 4884    |` cres 5007    Fn wfn 5589   ` cfv 5594  recscrecs 7053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-recs 7054
This theorem is referenced by:  rdgeq1  7089  rdgeq2  7090  dfoi  7948  oieq1  7949  oieq2  7950  ordtypecbv  7954  dfac12r  8538  zorn2g  8895  ttukey2g  8908  aomclem3  30930  aomclem8  30935
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