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Theorem freceq1 6219
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 5355 . . . . . . . . . 10  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( F `  (
g `  m )
)  =  ( G `
 ( g `  m ) ) )
32eleq2d 2169 . . . . . . . . 9  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( x  e.  ( F `  ( g `
 m ) )  <-> 
x  e.  ( G `
 ( g `  m ) ) ) )
43anbi2d 455 . . . . . . . 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 2397 . . . . . . 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 746 . . . . . 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 2217 . . . . 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 3958 . . . 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 6133 . . . 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 4754 . 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 6218 . 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 6218 . 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 2157 1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 670    = wceq 1299    e. wcel 1448   {cab 2086   E.wrex 2376   _Vcvv 2641   (/)c0 3310    |-> cmpt 3929   suc csuc 4225   omcom 4442   dom cdm 4477    |` cres 4479   ` cfv 5059  recscrecs 6131  freccfrec 6217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-res 4489  df-iota 5024  df-fv 5067  df-recs 6132  df-frec 6218
This theorem is referenced by:  frecuzrdgdom  10032  frecuzrdgfun  10034  frecuzrdgsuct  10038  seqeq1  10062  seqeq2  10063  seqeq3  10064  iseqvalcbv  10071  hashfz1  10370  ennnfonelemr  11728  ctinfom  11733  isomninn  12810
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