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Theorem rdgeq1 6440
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 5540 . . . . . 6  |-  ( F  =  G  ->  ( F `  ( g `  U. dom  g ) )  =  ( G `
 ( g `  U. dom  g ) ) )
21ifeq2d 3593 . . . . 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 3593 . . . 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 4123 . . 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 6405 . . 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 15 . 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 6439 . 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 6439 . 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 2353 1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   _Vcvv 2801   (/)c0 3468   ifcif 3578   U.cuni 3843    e. cmpt 4093   Lim wlim 4409   dom cdm 4705   ran crn 4706   ` cfv 5271  recscrecs 6403   reccrdg 6438
This theorem is referenced by:  rdgeq12  6442  rdgsucmpt2  6459  frsucmpt2  6468  seqomlem0  6477  omv  6527  oev  6529  dffi3  7200  hsmex  8074  axdc  8164  seqeq2  11066  seqval  11073  trpredlem1  24301  trpredtr  24304  trpredmintr  24305  valtar  25986  neibastop2  26413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-recs 6404  df-rdg 6439
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