Theorem resundir 5945

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

Assertion

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

Proof of Theorem resundir

Step	Hyp	Ref	Expression
1		indir 4237	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5631	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5631	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5631	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4117	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2762	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1540  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   × cxp 5617   ↾ cres 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-un 3908  df-in 3910  df-res 5631

This theorem is referenced by:  relresdm1  5984  imaundir  6099  fresaunres2  6696  fvunsn  7115  fvsnun1  7118  fvsnun2  7119  fsnunfv  7123  fsnunres  7124  frrlem12  8230  domss2  9053  axdc3lem4  10347  fseq1p1m1  13501  hashgval  14240  hashinf  14242  setsres  17089  setscom  17091  setsid  17118  pwssplit1  20963  nosupbnd2lem1  27625  noinfbnd2lem1  27640  noetasuplem2  27644  noetasuplem3  27645  noetasuplem4  27646  noetainflem2  27648  ex-res  30385  padct  32663  eulerpartlemt  34345  poimirlem3  37613  mapfzcons1  42700  diophrw  42742  eldioph2lem1  42743  eldioph2lem2  42744  diophin  42755  pwssplit4  43072