Theorem resundir 5989

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 4233	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5666	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5666	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5666	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4167	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2796	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3453   ∩ cin 4091   ∪ cun 4151   × cxp 5652   ↾ cres 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-in 4093  df-un 4153  df-res 5666

This theorem is used by:  relresdm1  6030  imaundir  6147  fresaunres2  6762  fvunsn  7191  fvsnun1  7194  fvsnun2  7195  fsnunfv  7199  fsnunres  7200  frrlem12  8326  domss2  9166  axdc3lem4  10541  fseq1p1m1  13742  hashgval  14487  hashinf  14489  setsres  17368  setscom  17370  setsid  17397  pwssplit1  21346  nosupbnd2lem1  28084  noinfbnd2lem1  28099  noetasuplem2  28103  noetasuplem3  28104  noetasuplem4  28105  noetainflem2  28107  ex-res  31054  padct  33364  eulerpartlemt  35056  poimirlem3  38681  ecunres  39466  mapfzcons1  43868  diophrw  43910  eldioph2lem1  43911  eldioph2lem2  43912  diophin  43923  pwssplit4  44231