Theorem resundir 5992

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 4266	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5677	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5677	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5677	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4146	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2767	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3463   ∪ cun 3929   ∩ cin 3930   × cxp 5663   ↾ cres 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-un 3936  df-in 3938  df-res 5677

This theorem is referenced by:  relresdm1  6031  imaundir  6150  fresaunres2  6760  fvunsn  7181  fvsnun1  7184  fvsnun2  7185  fsnunfv  7189  fsnunres  7190  frrlem12  8304  wfrlem14OLD  8344  domss2  9158  axdc3lem4  10475  fseq1p1m1  13620  hashgval  14354  hashinf  14356  setsres  17197  setscom  17199  setsid  17226  pwssplit1  21026  nosupbnd2lem1  27696  noinfbnd2lem1  27711  noetasuplem2  27715  noetasuplem3  27716  noetasuplem4  27717  noetainflem2  27719  ex-res  30388  padct  32666  eulerpartlemt  34332  poimirlem3  37589  mapfzcons1  42691  diophrw  42733  eldioph2lem1  42734  eldioph2lem2  42735  diophin  42746  pwssplit4  43064