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Theorem rdgeq2 6319
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 3254 . . . 4  |-  ( A  =  B  ->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )  =  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) )
21mpteq2dv 4055 . . 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 6253 . . 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 6317 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
6 df-irdg 6317 . 2  |-  rec ( F ,  B )  = recs ( ( g  e. 
_V  |->  ( B  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
74, 5, 63eqtr4g 2215 1  |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335   _Vcvv 2712    u. cun 3100   U_ciun 3849    |-> cmpt 4025   dom cdm 4586   ` cfv 5170  recscrecs 6251   reccrdg 6316
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-iota 5135  df-fv 5178  df-recs 6252  df-irdg 6317
This theorem is referenced by:  rdg0g  6335  oav  6401
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