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Theorem fixun 25473
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 3532 . . . 4  |-  ( ( A  u.  B )  i^i  _I  )  =  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
21dmeqi 5011 . . 3  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  dom  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
3 dmun 5016 . . 3  |-  dom  (
( A  i^i  _I  )  u.  ( B  i^i  _I  ) )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
42, 3eqtri 2407 . 2  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
5 df-fix 25424 . 2  |-  Fix ( A  u.  B )  =  dom  ( ( A  u.  B )  i^i 
_I  )
6 df-fix 25424 . . 3  |-  Fix A  =  dom  ( A  i^i  _I  )
7 df-fix 25424 . . 3  |-  Fix B  =  dom  ( B  i^i  _I  )
86, 7uneq12i 3442 . 2  |-  ( Fix A  u.  Fix B
)  =  ( dom  ( A  i^i  _I  )  u.  dom  ( B  i^i  _I  ) )
94, 5, 83eqtr4i 2417 1  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3261    i^i cin 3262    _I cid 4434   dom cdm 4818   Fixcfix 25402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-dm 4828  df-fix 25424
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