Theorem fixun 36420

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 4242	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∩ I ) = ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
2	1	dmeqi 5899	. . 3 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I ))
3		dmun 5905	. . 3 ⊢ dom ((𝐴 ∩ I ) ∪ (𝐵 ∩ I )) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
4	2, 3	eqtri 2789	. 2 ⊢ dom ((𝐴 ∪ 𝐵) ∩ I ) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
5		df-fix 36370	. 2 ⊢ Fix (𝐴 ∪ 𝐵) = dom ((𝐴 ∪ 𝐵) ∩ I )
6		df-fix 36370	. . 3 ⊢ Fix 𝐴 = dom (𝐴 ∩ I )
7		df-fix 36370	. . 3 ⊢ Fix 𝐵 = dom (𝐵 ∩ I )
8	6, 7	uneq12i 4123	. 2 ⊢ ( Fix 𝐴 ∪ Fix 𝐵) = (dom (𝐴 ∩ I ) ∪ dom (𝐵 ∩ I ))
9	4, 5, 8	3eqtr4i 2799	1 ⊢ Fix (𝐴 ∪ 𝐵) = ( Fix 𝐴 ∪ Fix 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3906   ∩ cin 3907   I cid 5560  dom cdm 5666   Fix cfix 36346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5676  df-fix 36370

This theorem is used by: (None)