Theorem resundir 2583

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 1678	. 2 ⊢ ((A ∪ B) ∩ (C × V)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
2		df-res 2430	. 2 ⊢ ((A ∪ B) ↾ C) = ((A ∪ B) ∩ (C × V))
3		df-res 2430	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
4		df-res 2430	. . 3 ⊢ (B ↾ C) = (B ∩ (C × V))
5	3, 4	uneq12i 1609	. 2 ⊢ ((A ↾ C) ∪ (B ↾ C)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
6	1, 2, 5	3eqtr4 1126	1 ⊢ ((A ∪ B) ↾ C) = ((A ↾ C) ∪ (B ↾ C))

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

This theorem is referenced by:  mapunen 3397  facnnt 4870  fac0 4871  ruclem6 4890  ruclem7 4891

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-res 2430

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