Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fixun Structured version   Visualization version   GIF version

Theorem fixun 31186
 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 3834 . . . 4 ((𝐴𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
21dmeqi 5247 . . 3 dom ((𝐴𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3 dmun 5253 . . 3 dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
42, 3eqtri 2632 . 2 dom ((𝐴𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5 df-fix 31135 . 2 Fix (𝐴𝐵) = dom ((𝐴𝐵) ∩ I )
6 df-fix 31135 . . 3 Fix 𝐴 = dom (𝐴 ∩ I )
7 df-fix 31135 . . 3 Fix 𝐵 = dom (𝐵 ∩ I )
86, 7uneq12i 3727 . 2 ( Fix 𝐴 Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
94, 5, 83eqtr4i 2642 1 Fix (𝐴𝐵) = ( Fix 𝐴 Fix 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∪ cun 3538   ∩ cin 3539   I cid 4948  dom cdm 5038   Fix cfix 31111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-dm 5048  df-fix 31135 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator