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Theorem recseq 5952
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 5205 . . . . . . . 8  |-  ( F  =  G  ->  ( F `  ( a  |`  c ) )  =  ( G `  (
a  |`  c ) ) )
21eqeq2d 2067 . . . . . . 7  |-  ( F  =  G  ->  (
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
32ralbidv 2343 . . . . . 6  |-  ( F  =  G  ->  ( A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )  <->  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) )
43anbi2d 445 . . . . 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 2344 . . . 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 2171 . . 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 3619 . 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 5951 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
9 df-recs 5951 . 2  |- recs ( G )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
107, 8, 93eqtr4g 2113 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   {cab 2042   A.wral 2323   E.wrex 2324   U.cuni 3608   Oncon0 4128    |` cres 4375    Fn wfn 4925   ` cfv 4930  recscrecs 5950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-recs 5951
This theorem is referenced by:  rdgeq1  5989  rdgeq2  5990  freceq1  6010  freceq2  6011  frecsuclem1  6018  frecsuclem2  6020
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