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Theorem rdgeq2 7130
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 3885 . . . 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 4490 . . 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 7092 . . 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 17 . 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 7128 . 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 7128 . 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 2510 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444   _Vcvv 3045   (/)c0 3731   ifcif 3881   U.cuni 4198    |-> cmpt 4461   dom cdm 4834   ran crn 4835   Lim wlim 5424   ` cfv 5582  recscrecs 7089   reccrdg 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-iota 5546  df-fv 5590  df-wrecs 7028  df-recs 7090  df-rdg 7128
This theorem is referenced by:  rdgeq12  7131  rdg0g  7145  oav  7213  itunifval  8846  hsmex  8862  ltweuz  12175  seqeq1  12216  dfrdg2  30442  trpredeq3  30463  finxpeq2  31779  finxpreclem6  31788  finxpsuclem  31789
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