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Theorem fixun 32529
Description: The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
fixun Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)

Proof of Theorem fixun
StepHypRef Expression
1 indir 4076 . . . 4 ((𝐴𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
21dmeqi 5528 . . 3 dom ((𝐴𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3 dmun 5534 . . 3 dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
42, 3eqtri 2821 . 2 dom ((𝐴𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5 df-fix 32479 . 2 Fix (𝐴𝐵) = dom ((𝐴𝐵) ∩ I )
6 df-fix 32479 . . 3 Fix 𝐴 = dom (𝐴 ∩ I )
7 df-fix 32479 . . 3 Fix 𝐵 = dom (𝐵 ∩ I )
86, 7uneq12i 3963 . 2 ( Fix 𝐴 Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
94, 5, 83eqtr4i 2831 1 Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cun 3767  cin 3768   I cid 5219  dom cdm 5312   Fix cfix 32455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-dm 5322  df-fix 32479
This theorem is referenced by: (None)
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