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Theorem recseq 7045
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 5855 . . . . . . . 8  |-  ( F  =  G  ->  ( F `  ( a  |`  c ) )  =  ( G `  (
a  |`  c ) ) )
21eqeq2d 2457 . . . . . . 7  |-  ( F  =  G  ->  (
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
32ralbidv 2882 . . . . . 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 2954 . . . 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 2579 . . 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 4244 . 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 7044 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
9 df-recs 7044 . 2  |- recs ( G )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
107, 8, 93eqtr4g 2509 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383   {cab 2428   A.wral 2793   E.wrex 2794   U.cuni 4234   Oncon0 4868    |` cres 4991    Fn wfn 5573   ` cfv 5578  recscrecs 7043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-recs 7044
This theorem is referenced by:  rdgeq1  7079  rdgeq2  7080  dfoi  7939  oieq1  7940  oieq2  7941  ordtypecbv  7945  dfac12r  8529  zorn2g  8886  ttukey2g  8899  aomclem3  30977  aomclem8  30982
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