Theorem resundi 3435

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 3283	. . . 4 ⊢ ((B ∪ C) × V) = ((B × V) ∪ (C × V))
2	1	ineq2i 2265	. . 3 ⊢ (A ∩ ((B ∪ C) × V)) = (A ∩ ((B × V) ∪ (C × V)))
3		indi 2302	. . 3 ⊢ (A ∩ ((B × V) ∪ (C × V))) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
4	2, 3	eqtri 1542	. 2 ⊢ (A ∩ ((B ∪ C) × V)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
5		df-res 3247	. 2 ⊢ (A ↾ (B ∪ C)) = (A ∩ ((B ∪ C) × V))
6		df-res 3247	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
7		df-res 3247	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
8	6, 7	uneq12i 2233	. 2 ⊢ ((A ↾ B) ∪ (A ↾ C)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
9	4, 5, 8	3eqtr4i 1552	1 ⊢ (A ↾ (B ∪ C)) = ((A ↾ B) ∪ (A ↾ C))

Syntax hints:   = wceq 997  Vcvv 1858   ∪ cun 2096   ∩ cin 2097   × cxp 3225   ↾ cres 3229

This theorem is referenced by:  imaun 3517  mapunen 4567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-v 1859  df-un 2101  df-in 2102  df-opab 2722  df-xp 3241  df-res 3247

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