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Theorem freceq1 6257
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 108 . . . . . . . . . . 11  |-  ( ( F  =  G  /\  g  e.  _V )  ->  F  =  G )
21fveq1d 5391 . . . . . . . . . 10  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( F `  (
g `  m )
)  =  ( G `
 ( g `  m ) ) )
32eleq2d 2187 . . . . . . . . 9  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( x  e.  ( F `  ( g `
 m ) )  <-> 
x  e.  ( G `
 ( g `  m ) ) ) )
43anbi2d 459 . . . . . . . 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 2415 . . . . . . 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 765 . . . . . 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 2235 . . . . 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 3988 . . . 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 6171 . . . 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 4788 . 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 6256 . 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 6256 . 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 2175 1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660   (/)c0 3333    |-> cmpt 3959   suc csuc 4257   omcom 4474   dom cdm 4509    |` cres 4511   ` cfv 5093  recscrecs 6169  freccfrec 6255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-res 4521  df-iota 5058  df-fv 5101  df-recs 6170  df-frec 6256
This theorem is referenced by:  frecuzrdgdom  10159  frecuzrdgfun  10161  frecuzrdgsuct  10165  seqeq1  10189  seqeq2  10190  seqeq3  10191  iseqvalcbv  10198  hashfz1  10497  ennnfonelemr  11863  ctinfom  11868  isomninn  13153
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