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Theorem rdgeq2 6370
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 variables  x  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3282 . . . 4  |-  ( A  =  B  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) )
21mpteq2dv 4093 . . 3  |-  ( A  =  B  ->  (
g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  =  ( g  e.  _V  |->  ( B  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) ) )
3 recseq 6304 . . 3  |-  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  =  ( g  e.  _V  |->  ( B  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) )  -> recs ( (
g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )  = recs (
( g  e.  _V  |->  ( B  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) ) )
42, 3syl 14 . 2  |-  ( A  =  B  -> recs ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )  = recs (
( g  e.  _V  |->  ( B  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) ) )
5 df-irdg 6368 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
6 df-irdg 6368 . 2  |-  rec ( F ,  B )  = recs ( ( g  e. 
_V  |->  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
74, 5, 63eqtr4g 2235 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   _Vcvv 2737    u. cun 3127   U_ciun 3886    |-> cmpt 4063   dom cdm 4625   ` cfv 5215  recscrecs 6302   reccrdg 6367
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-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-iota 5177  df-fv 5223  df-recs 6303  df-irdg 6368
This theorem is referenced by:  rdg0g  6386  oav  6452
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