Theorem resundir 3385

Description: Distributive law for restriction over union.

Assertion

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

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 2256	. 2 ⊢ ((A ∪ B) ∩ (C × V)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
2		df-res 3196	. 2 ⊢ ((A ∪ B) ↾ C) = ((A ∪ B) ∩ (C × V))
3		df-res 3196	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
4		df-res 3196	. . 3 ⊢ (B ↾ C) = (B ∩ (C × V))
5	3, 4	uneq12i 2185	. 2 ⊢ ((A ↾ C) ∪ (B ↾ C)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
6	1, 2, 5	3eqtr4 1508	1 ⊢ ((A ∪ B) ↾ C) = ((A ↾ C) ∪ (B ↾ C))

Syntax hints:   = wceq 958  Vcvv 1814   ∪ cun 2048   ∩ cin 2049   × cxp 3174   ↾ cres 3178

This theorem is referenced by:  imaun2 3467  fvsnun1 3801  fvsnun2 3802  mapunen 4508  acdc2lem2 7490  acdc5lem2 7493  ruclem6 7516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-res 3196

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