Theorem fixun 35892

Description: The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fixun	⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)

Proof of Theorem fixun

Step	Hyp	Ref	Expression
1		indir 4251	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
2	1	dmeqi 5870	. . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3		dmun 5876	. . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
4	2, 3	eqtri 2753	. 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5		df-fix 35842	. 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I )
6		df-fix 35842	. . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
7		df-fix 35842	. . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I )
8	6, 7	uneq12i 4131	. 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
9	4, 5, 8	3eqtr4i 2763	1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)

Syntax hints:   = wceq 1540   ∪ cun 3914   ∩ cin 3915   I cid 5534  dom cdm 5640   Fix cfix 35818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-dm 5650  df-fix 35842

This theorem is referenced by: (None)