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Theorem freceq1 6113
Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
freceq1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )

Proof of Theorem freceq1
Dummy variables  x  g  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . . . . . . . . . 11  |-  ( ( F  =  G  /\  g  e.  _V )  ->  F  =  G )
21fveq1d 5272 . . . . . . . . . 10  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( F `  (
g `  m )
)  =  ( G `
 ( g `  m ) ) )
32eleq2d 2154 . . . . . . . . 9  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( x  e.  ( F `  ( g `
 m ) )  <-> 
x  e.  ( G `
 ( g `  m ) ) ) )
43anbi2d 452 . . . . . . . 8  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  <->  ( dom  g  =  suc  m  /\  x  e.  ( G `  ( g `  m
) ) ) ) )
54rexbidv 2377 . . . . . . 7  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  <->  E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) ) ) )
65orbi1d 738 . . . . . 6  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) )  <->  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) ) )
76abbidv 2202 . . . . 5  |-  ( ( F  =  G  /\  g  e.  _V )  ->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) }  =  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )
87mpteq2dva 3905 . . . 4  |-  ( F  =  G  ->  (
g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )
9 recseq 6027 . . . 4  |-  ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  =  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } )  -> recs ( (
g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) )
108, 9syl 14 . . 3  |-  ( F  =  G  -> recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  = recs (
( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) ) )
1110reseq1d 4682 . 2  |-  ( F  =  G  ->  (recs ( ( g  e. 
_V  |->  { x  |  ( E. m  e. 
om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
)
12 df-frec 6112 . 2  |- frec ( F ,  A )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
13 df-frec 6112 . 2  |- frec ( G ,  A )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( G `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  A ) ) } ) )  |`  om )
1411, 12, 133eqtr4g 2142 1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1287    e. wcel 1436   {cab 2071   E.wrex 2356   _Vcvv 2615   (/)c0 3275    |-> cmpt 3876   suc csuc 4168   omcom 4380   dom cdm 4413    |` cres 4415   ` cfv 4983  recscrecs 6025  freccfrec 6111
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-res 4425  df-iota 4948  df-fv 4991  df-recs 6026  df-frec 6112
This theorem is referenced by:  frecuzrdgdom  9756  frecuzrdgfun  9758  frecuzrdgsuct  9762  iseqeq1  9788  iseqeq2  9789  iseqeq3  9790  iseqeq4  9791  iseqval  9802  iseqvalcbv  9803  hashfz1  10109
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