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Theorem fixun 25719
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 3581 . . . 4  |-  ( ( A  u.  B )  i^i  _I  )  =  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
21dmeqi 5063 . . 3  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  dom  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
3 dmun 5068 . . 3  |-  dom  (
( A  i^i  _I  )  u.  ( B  i^i  _I  ) )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
42, 3eqtri 2455 . 2  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
5 df-fix 25668 . 2  |-  Fix ( A  u.  B )  =  dom  ( ( A  u.  B )  i^i 
_I  )
6 df-fix 25668 . . 3  |-  Fix A  =  dom  ( A  i^i  _I  )
7 df-fix 25668 . . 3  |-  Fix B  =  dom  ( B  i^i  _I  )
86, 7uneq12i 3491 . 2  |-  ( Fix A  u.  Fix B
)  =  ( dom  ( A  i^i  _I  )  u.  dom  ( B  i^i  _I  ) )
94, 5, 83eqtr4i 2465 1  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3310    i^i cin 3311    _I cid 4485   dom cdm 4870   Fixcfix 25644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-dm 4880  df-fix 25668
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