Theorem resundi 2582

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65.

Assertion

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

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 2462	. . . 4 ⊢ ((B ∪ C) × V) = ((B × V) ∪ (C × V))
2	1	ineq2i 1642	. . 3 ⊢ (A ∩ ((B ∪ C) × V)) = (A ∩ ((B × V) ∪ (C × V)))
3		indi 1676	. . 3 ⊢ (A ∩ ((B × V) ∪ (C × V))) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
4	2, 3	eqtr 1119	. 2 ⊢ (A ∩ ((B ∪ C) × V)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
5		df-res 2430	. 2 ⊢ (A ↾ (B ∪ C)) = (A ∩ ((B ∪ C) × V))
6		df-res 2430	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
7		df-res 2430	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
8	6, 7	uneq12i 1609	. 2 ⊢ ((A ↾ B) ∪ (A ↾ C)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
9	4, 5, 8	3eqtr4 1126	1 ⊢ (A ↾ (B ∪ C)) = ((A ↾ B) ∪ (A ↾ C))

Syntax hints:   = wceq 1091  Vcvv 1348   ∪ cun 1485   ∩ cin 1486   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  imaun 2647  mapunen 3397

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-opab 2098  df-xp 2424  df-res 2430

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