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Theorem fixun 24449
Description: The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
fixun  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )

Proof of Theorem fixun
StepHypRef Expression
1 indir 3417 . . . 4  |-  ( ( A  u.  B )  i^i  _I  )  =  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
21dmeqi 4880 . . 3  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  dom  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
3 dmun 4885 . . 3  |-  dom  (
( A  i^i  _I  )  u.  ( B  i^i  _I  ) )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
42, 3eqtri 2303 . 2  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
5 df-fix 24400 . 2  |-  Fix ( A  u.  B )  =  dom  ( ( A  u.  B )  i^i 
_I  )
6 df-fix 24400 . . 3  |-  Fix A  =  dom  ( A  i^i  _I  )
7 df-fix 24400 . . 3  |-  Fix B  =  dom  ( B  i^i  _I  )
86, 7uneq12i 3327 . 2  |-  ( Fix A  u.  Fix B
)  =  ( dom  ( A  i^i  _I  )  u.  dom  ( B  i^i  _I  ) )
94, 5, 83eqtr4i 2313 1  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150    i^i cin 3151    _I cid 4304   dom cdm 4689   Fixcfix 24378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699  df-fix 24400
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