Theorem resundir 4983

Description: Distributive law for restriction over union. (Contributed by set.mm contributors, 23-Sep-2004.)

Assertion

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

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 3504	. 2 ⊢ ((A ∪ B) ∩ (C × V)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
2		df-res 4789	. 2 ⊢ ((A ∪ B) ↾ C) = ((A ∪ B) ∩ (C × V))
3		df-res 4789	. . 3 ⊢ (A ↾ C) = (A ∩ (C × V))
4		df-res 4789	. . 3 ⊢ (B ↾ C) = (B ∩ (C × V))
5	3, 4	uneq12i 3417	. 2 ⊢ ((A ↾ C) ∪ (B ↾ C)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
6	1, 2, 5	3eqtr4i 2383	1 ⊢ ((A ∪ B) ↾ C) = ((A ↾ C) ∪ (B ↾ C))

Syntax hints:   = wceq 1642  Vcvv 2860   ∪ cun 3208   ∩ cin 3209   × cxp 4771   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-res 4789

This theorem is referenced by:  imaundir  5041  fvunsn  5445  fvsnun1  5448  fvsnun2  5449