Theorem resundi 4981

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by set.mm contributors, 30-Sep-2002.)

Assertion

Ref	Expression
resundi	⊢ (A ↾ (B ∪ C)) = ((A ↾ B) ∪ (A ↾ C))

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 4833	. . . 4 ⊢ ((B ∪ C) × V) = ((B × V) ∪ (C × V))
2	1	ineq2i 3454	. . 3 ⊢ (A ∩ ((B ∪ C) × V)) = (A ∩ ((B × V) ∪ (C × V)))
3		indi 3501	. . 3 ⊢ (A ∩ ((B × V) ∪ (C × V))) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
4	2, 3	eqtri 2373	. 2 ⊢ (A ∩ ((B ∪ C) × V)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
5		df-res 4788	. 2 ⊢ (A ↾ (B ∪ C)) = (A ∩ ((B ∪ C) × V))
6		df-res 4788	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
7		df-res 4788	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
8	6, 7	uneq12i 3416	. 2 ⊢ ((A ↾ B) ∪ (A ↾ C)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
9	4, 5, 8	3eqtr4i 2383	1 ⊢ (A ↾ (B ∪ C)) = ((A ↾ B) ∪ (A ↾ C))

Syntax hints:   = wceq 1642  Vcvv 2859   ∪ cun 3207   ∩ cin 3208   × cxp 4770   ↾ cres 4774

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-opab 4623  df-xp 4784  df-res 4788

This theorem is referenced by:  imaundi  5039