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Theorem rdgeq2 6441
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
rdgeq2  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )

Proof of Theorem rdgeq2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 ifeq1 3582 . . . 4  |-  ( A  =  B  ->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) )  =  if ( g  =  (/) ,  B ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) )
21mpteq2dv 4123 . . 3  |-  ( A  =  B  ->  (
g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) )  =  ( g  e.  _V  |->  if ( g  =  (/) ,  B ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )
3 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  =  (/) ,  B ,  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 ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )  = recs (
( g  e.  _V  |->  if ( g  =  (/) ,  B ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) ) )
42, 3syl 15 . 2  |-  ( A  =  B  -> 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  =  (/) ,  B ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) ) )
5 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
) ) ) ) ) )
6 df-rdg 6439 . 2  |-  rec ( F ,  B )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  B ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  (
g `  U. dom  g
) ) ) ) ) )
74, 5, 63eqtr4g 2353 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
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  rdg0g  6456  oav  6526  itunifval  8058  hsmex  8074  ltweuz  11040  seqeq1  11065  dfrdg2  24223  trpredeq3  24296  valtar  25986  trclval  25997
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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