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Theorem fixun 24520
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 3430 . . . 4  |-  ( ( A  u.  B )  i^i  _I  )  =  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
21dmeqi 4896 . . 3  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  dom  ( ( A  i^i  _I  )  u.  ( B  i^i  _I  ) )
3 dmun 4901 . . 3  |-  dom  (
( A  i^i  _I  )  u.  ( B  i^i  _I  ) )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
42, 3eqtri 2316 . 2  |-  dom  (
( A  u.  B
)  i^i  _I  )  =  ( dom  ( A  i^i  _I  )  u. 
dom  ( B  i^i  _I  ) )
5 df-fix 24471 . 2  |-  Fix ( A  u.  B )  =  dom  ( ( A  u.  B )  i^i 
_I  )
6 df-fix 24471 . . 3  |-  Fix A  =  dom  ( A  i^i  _I  )
7 df-fix 24471 . . 3  |-  Fix B  =  dom  ( B  i^i  _I  )
86, 7uneq12i 3340 . 2  |-  ( Fix A  u.  Fix B
)  =  ( dom  ( A  i^i  _I  )  u.  dom  ( B  i^i  _I  ) )
94, 5, 83eqtr4i 2326 1  |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163    i^i cin 3164    _I cid 4320   dom cdm 4705   Fixcfix 24449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715  df-fix 24471
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