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Theorem rdgeq1 7074
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )

Proof of Theorem rdgeq1
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 fveq1 5863 . . . . . 6  |-  ( F  =  G  ->  ( F `  ( g `  U. dom  g ) )  =  ( G `
 ( g `  U. dom  g ) ) )
21ifeq2d 3958 . . . . 5  |-  ( F  =  G  ->  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )  =  if ( Lim  dom  g ,  U. ran  g ,  ( G `  ( g `  U. dom  g ) ) ) )
32ifeq2d 3958 . . . 4  |-  ( F  =  G  ->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )  =  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  (
g `  U. dom  g
) ) ) ) )
43mpteq2dv 4534 . . 3  |-  ( F  =  G  ->  (
g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )  =  ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  ( g `  U. dom  g ) ) ) ) ) )
5 recseq 7040 . . 3  |-  ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )  =  ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  ( g `  U. dom  g ) ) ) ) )  -> recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )  = recs (
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  ( g `
 U. dom  g
) ) ) ) ) ) )
64, 5syl 16 . 2  |-  ( F  =  G  -> recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )  = recs (
( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  ( g `
 U. dom  g
) ) ) ) ) ) )
7 df-rdg 7073 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
8 df-rdg 7073 . 2  |-  rec ( G ,  A )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( G `  (
g `  U. dom  g
) ) ) ) ) )
96, 7, 83eqtr4g 2533 1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   _Vcvv 3113   (/)c0 3785   ifcif 3939   U.cuni 4245    |-> cmpt 4505   Lim wlim 4879   dom cdm 4999   ran crn 5000   ` cfv 5586  recscrecs 7038   reccrdg 7072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-un 3481  df-if 3940  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-iota 5549  df-fv 5594  df-recs 7039  df-rdg 7073
This theorem is referenced by:  rdgeq12  7076  rdgsucmpt2  7093  frsucmpt2  7102  seqomlem0  7111  omv  7159  oev  7161  dffi3  7887  hsmex  8808  axdc  8897  seqeq2  12074  seqval  12081  trpredlem1  28884  trpredtr  28887  trpredmintr  28888  neibastop2  29780
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