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Theorem fixun 29486
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 3751 . . . 4  |-  ( ( A  u.  B )  i^i  _I  )  =  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
21dmeqi 5210 . . 3  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  dom  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
3 dmun 5215 . . 3  |-  dom  (
( A  i^i  _I  )  u.  ( B  i^i  _I  ) )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
42, 3eqtri 2496 . 2  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
5 df-fix 29435 . 2  |-  Fix ( A  u.  B )  =  dom  ( ( A  u.  B )  i^i 
_I  )
6 df-fix 29435 . . 3  |-  Fix A  =  dom  ( A  i^i  _I  )
7 df-fix 29435 . . 3  |-  Fix B  =  dom  ( B  i^i  _I  )
86, 7uneq12i 3661 . 2  |-  ( Fix A  u.  Fix B
)  =  ( dom  ( A  i^i  _I  )  u.  dom  ( B  i^i  _I  ) )
94, 5, 83eqtr4i 2506 1  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    u. cun 3479    i^i cin 3480    _I cid 4796   dom cdm 5005   Fixcfix 29411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-dm 5015  df-fix 29435
This theorem is referenced by: (None)
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