Theorem resundir 5994

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 4247	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5674	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5674	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5674	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4128	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2802	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1567  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   × cxp 5660   ↾ cres 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-res 5674

This theorem is referenced by:  relresdm1  6036  imaundir  6149  fresaunres2  6751  fvunsn  7178  fvsnun1  7181  fvsnun2  7182  fsnunfv  7186  fsnunres  7187  frrlem12  8294  domss2  9124  axdc3lem4  10437  fseq1p1m1  13626  hashgval  14369  hashinf  14371  setsres  17238  setscom  17240  setsid  17267  pwssplit1  21158  nosupbnd2lem1  27845  noinfbnd2lem1  27860  noetasuplem2  27864  noetasuplem3  27865  noetasuplem4  27866  noetainflem2  27868  ex-res  30733  padct  33004  eulerpartlemt  34706  poimirlem3  38162  ecunres  38933  mapfzcons1  43340  diophrw  43382  eldioph2lem1  43383  eldioph2lem2  43384  diophin  43395  pwssplit4  43708