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Theorem rdgeq2 6351
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 3274 . . . 4  |-  ( A  =  B  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) )
21mpteq2dv 4080 . . 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 6285 . . 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 6349 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
6 df-irdg 6349 . 2  |-  rec ( F ,  B )  = recs ( ( g  e. 
_V  |->  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
74, 5, 63eqtr4g 2228 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   _Vcvv 2730    u. cun 3119   U_ciun 3873    |-> cmpt 4050   dom cdm 4611   ` cfv 5198  recscrecs 6283   reccrdg 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-iota 5160  df-fv 5206  df-recs 6284  df-irdg 6349
This theorem is referenced by:  rdg0g  6367  oav  6433
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