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Theorem rdgeq1 6371
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 5514 . . . . . 6  |-  ( F  =  G  ->  ( F `  ( g `  x ) )  =  ( G `  (
g `  x )
) )
21iuneq2d 3911 . . . . 5  |-  ( F  =  G  ->  U_ x  e.  dom  g ( F `
 ( g `  x ) )  = 
U_ x  e.  dom  g ( G `  ( g `  x
) ) )
32uneq2d 3289 . . . 4  |-  ( F  =  G  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( A  u.  U_ x  e.  dom  g
( G `  (
g `  x )
) ) )
43mpteq2dv 4094 . . 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 6306 . . 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 6370 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
8 df-irdg 6370 . 2  |-  rec ( G ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( G `  (
g `  x )
) ) ) )
96, 7, 83eqtr4g 2235 1  |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   _Vcvv 2737    u. cun 3127   U_ciun 3886    |-> cmpt 4064   dom cdm 4626   ` cfv 5216  recscrecs 6304   reccrdg 6369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-iota 5178  df-fv 5224  df-recs 6305  df-irdg 6370
This theorem is referenced by:  omv  6455  oeiv  6456
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