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Theorem rdgeq1 6671
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 5729 . . . . . 6  |-  ( F  =  G  ->  ( F `  ( g `  U. dom  g ) )  =  ( G `
 ( g `  U. dom  g ) ) )
21ifeq2d 3756 . . . . 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 3756 . . . 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 4298 . . 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 6636 . . 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 6670 . 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 6670 . 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 2495 1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   (/)c0 3630   ifcif 3741   U.cuni 4017    e. cmpt 4268   Lim wlim 4584   dom cdm 4880   ran crn 4881   ` cfv 5456  recscrecs 6634   reccrdg 6669
This theorem is referenced by:  rdgeq12  6673  rdgsucmpt2  6690  frsucmpt2  6699  seqomlem0  6708  omv  6758  oev  6760  dffi3  7438  hsmex  8314  axdc  8403  seqeq2  11329  seqval  11336  trpredlem1  25507  trpredtr  25510  trpredmintr  25511  neibastop2  26392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-un 3327  df-if 3742  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-iota 5420  df-fv 5464  df-recs 6635  df-rdg 6670
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