Theorem fixun 36101

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 4238	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
2	1	dmeqi 5853	. . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3		dmun 5859	. . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
4	2, 3	eqtri 2759	. 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5		df-fix 36051	. 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I )
6		df-fix 36051	. . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
7		df-fix 36051	. . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I )
8	6, 7	uneq12i 4118	. 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
9	4, 5, 8	3eqtr4i 2769	1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)

Syntax hints:   = wceq 1541   ∪ cun 3899   ∩ cin 3900   I cid 5518  dom cdm 5624   Fix cfix 36027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-dm 5634  df-fix 36051

This theorem is referenced by: (None)