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Theorem rdgeq1 6068
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 variables  x  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5252 . . . . . 6  |-  ( F  =  G  ->  ( F `  ( g `  x ) )  =  ( G `  (
g `  x )
) )
21iuneq2d 3729 . . . . 5  |-  ( F  =  G  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  g ( G `  ( g `  x
) ) )
32uneq2d 3138 . . . 4  |-  ( F  =  G  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  g
( G `  (
g `  x )
) ) )
43mpteq2dv 3895 . . 3  |-  ( F  =  G  ->  (
g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  =  ( g  e.  _V  |->  ( A  u.  U_ x  e. 
dom  g ( G `
 ( g `  x ) ) ) ) )
5 recseq 6003 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  =  ( g  e.  _V  |->  ( A  u.  U_ x  e. 
dom  g ( G `
 ( g `  x ) ) ) )  -> recs ( (
g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )  = recs (
( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( G `
 ( g `  x ) ) ) ) ) )
64, 5syl 14 . 2  |-  ( F  =  G  -> recs ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )  = recs (
( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( G `
 ( g `  x ) ) ) ) ) )
7 df-irdg 6067 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
8 df-irdg 6067 . 2  |-  rec ( G ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( G `  (
g `  x )
) ) ) )
96, 7, 83eqtr4g 2140 1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2612    u. cun 2982   U_ciun 3704    |-> cmpt 3865   dom cdm 4401   ` cfv 4969  recscrecs 6001   reccrdg 6066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-iota 4934  df-fv 4977  df-recs 6002  df-irdg 6067
This theorem is referenced by:  omv  6148  oeiv  6149
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