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Theorem recseq 6053
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 5288 . . . . . . . 8  |-  ( F  =  G  ->  ( F `  ( a  |`  c ) )  =  ( G `  (
a  |`  c ) ) )
21eqeq2d 2099 . . . . . . 7  |-  ( F  =  G  ->  (
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
32ralbidv 2380 . . . . . 6  |-  ( F  =  G  ->  ( A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
43anbi2d 452 . . . . 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 2381 . . . 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 2205 . . 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 3659 . 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 6052 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
9 df-recs 6052 . 2  |- recs ( G )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
107, 8, 93eqtr4g 2145 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   {cab 2074   A.wral 2359   E.wrex 2360   U.cuni 3648   Oncon0 4181    |` cres 4430    Fn wfn 4997   ` cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-recs 6052
This theorem is referenced by:  rdgeq1  6118  rdgeq2  6119  freceq1  6139  freceq2  6140  frecsuclem  6153
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