Theorem fixun 34138

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 4206	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
2	1	dmeqi 5802	. . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3		dmun 5808	. . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
4	2, 3	eqtri 2766	. 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5		df-fix 34088	. 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I )
6		df-fix 34088	. . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
7		df-fix 34088	. . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I )
8	6, 7	uneq12i 4091	. 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
9	4, 5, 8	3eqtr4i 2776	1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)

Syntax hints:   = wceq 1539   ∪ cun 3881   ∩ cin 3882   I cid 5479  dom cdm 5580   Fix cfix 34064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-dm 5590  df-fix 34088

This theorem is referenced by: (None)