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Theorem rdgeq2 6085
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 3136 . . . 4  |-  ( A  =  B  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) )
21mpteq2dv 3904 . . 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 6019 . . 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 6083 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
6 df-irdg 6083 . 2  |-  rec ( F ,  B )  = recs ( ( g  e. 
_V  |->  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
74, 5, 63eqtr4g 2142 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   _Vcvv 2615    u. cun 2986   U_ciun 3713    |-> cmpt 3874   dom cdm 4410   ` cfv 4978  recscrecs 6017   reccrdg 6082
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-iota 4943  df-fv 4986  df-recs 6018  df-irdg 6083
This theorem is referenced by:  rdg0g  6101  oav  6163
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