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Theorem resundir 5855
Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
resundir ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem resundir
StepHypRef Expression
1 indir 4237 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2 df-res 5554 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 5554 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 5554 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4uneq12i 4123 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2857 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3480  cun 3917  cin 3918   × cxp 5540  cres 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-res 5554
This theorem is referenced by:  imaundir  5996  fresaunres2  6540  fvunsn  6932  fvsnun1  6935  fvsnun2  6936  fsnunfv  6940  fsnunres  6941  wfrlem14  7964  domss2  8673  axdc3lem4  9873  fseq1p1m1  12985  hashgval  13698  hashinf  13700  setsres  16525  setscom  16527  setsid  16538  pwssplit1  19831  ex-res  28229  funresdm1  30366  padct  30466  eulerpartlemt  31686  frrlem12  33191  nosupbnd2lem1  33272  noetalem2  33275  noetalem3  33276  poimirlem3  35005  mapfzcons1  39574  diophrw  39616  eldioph2lem1  39617  eldioph2lem2  39618  diophin  39629  pwssplit4  39949
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