Theorem resundir 5906

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 4209	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2		df-res 5601	. 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V))
3		df-res 5601	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5601	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	uneq12i 4095	. 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2776	1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3432   ∪ cun 3885   ∩ cin 3886   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-un 3892  df-in 3894  df-res 5601

This theorem is referenced by:  imaundir  6054  fresaunres2  6646  fvunsn  7051  fvsnun1  7054  fvsnun2  7055  fsnunfv  7059  fsnunres  7060  frrlem12  8113  wfrlem14OLD  8153  domss2  8923  axdc3lem4  10209  fseq1p1m1  13330  hashgval  14047  hashinf  14049  setsres  16879  setscom  16881  setsid  16909  pwssplit1  20321  ex-res  28805  funresdm1  30944  padct  31054  eulerpartlemt  32338  nosupbnd2lem1  33918  noinfbnd2lem1  33933  noetasuplem2  33937  noetasuplem3  33938  noetasuplem4  33939  noetainflem2  33941  poimirlem3  35780  mapfzcons1  40539  diophrw  40581  eldioph2lem1  40582  eldioph2lem2  40583  diophin  40594  pwssplit4  40914