ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  freceq1 Unicode version

Theorem freceq1 6089
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 5255 . . . . . . . . . 10  |-  ( ( F  =  G  /\  g  e.  _V )  ->  ( F `  (
g `  m )
)  =  ( G `
 ( g `  m ) ) )
32eleq2d 2152 . . . . . . . . 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 2375 . . . . . . 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 2200 . . . . 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 3894 . . . 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 6003 . . . 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 4670 . 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 6088 . 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 6088 . 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 2140 1  |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434   {cab 2069   E.wrex 2354   _Vcvv 2612   (/)c0 3269    |-> cmpt 3865   suc csuc 4156   omcom 4368   dom cdm 4401    |` cres 4403   ` cfv 4969  recscrecs 6001  freccfrec 6087
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-res 4413  df-iota 4934  df-fv 4977  df-recs 6002  df-frec 6088
This theorem is referenced by:  frecuzrdgdom  9714  frecuzrdgfun  9716  frecuzrdgsuct  9720  iseqeq1  9743  iseqeq2  9744  iseqeq3  9745  iseqeq4  9746  iseqval  9749  iseqvalcbv  9750  hashfz1  10026
  Copyright terms: Public domain W3C validator