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Theorem rdgeq2 6425
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 3569 . . . 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 4107 . . 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 6389 . . 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 6423 . 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 6423 . 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 2340 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   _Vcvv 2788   (/)c0 3455   ifcif 3565   U.cuni 3827    e. cmpt 4077   Lim wlim 4393   dom cdm 4689   ran crn 4690   ` cfv 5255  recscrecs 6387   reccrdg 6422
This theorem is referenced by:  rdgeq12  6426  rdg0g  6440  oav  6510  itunifval  8042  hsmex  8058  ltweuz  11024  seqeq1  11049  dfrdg2  24152  trpredeq3  24225  valtar  25883  trclval  25894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-un 3157  df-if 3566  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-recs 6388  df-rdg 6423
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