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Theorem freceq1 6407
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 109 . . . . . . . . . . 11  |-  ( ( F  =  G  /\  g  e.  _V )  ->  F  =  G )
21fveq1d 5529 . . . . . . . . . 10  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( F `  (
g `  m )
)  =  ( G `
 ( g `  m ) ) )
32eleq2d 2257 . . . . . . . . 9  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( x  e.  ( F `  ( g `
 m ) )  <-> 
x  e.  ( G `
 ( g `  m ) ) ) )
43anbi2d 464 . . . . . . . 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 2488 . . . . . . 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 792 . . . . . 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 2305 . . . . 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 4105 . . . 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 6321 . . . 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 4918 . 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 6406 . 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 6406 . 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 2245 1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709    = wceq 1363    e. wcel 2158   {cab 2173   E.wrex 2466   _Vcvv 2749   (/)c0 3434    |-> cmpt 4076   suc csuc 4377   omcom 4601   dom cdm 4638    |` cres 4640   ` cfv 5228  recscrecs 6319  freccfrec 6405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-res 4650  df-iota 5190  df-fv 5236  df-recs 6320  df-frec 6406
This theorem is referenced by:  frecuzrdgdom  10432  frecuzrdgfun  10434  frecuzrdgsuct  10438  seqeq1  10462  seqeq2  10463  seqeq3  10464  iseqvalcbv  10471  hashfz1  10777  ennnfonelemr  12438  ctinfom  12443  isomninn  15133  iswomninn  15152  ismkvnn  15155
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