ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resundir GIF version

Theorem resundir 4905
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 3376 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2 df-res 4623 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4623 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4623 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4uneq12i 3279 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2201 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1348  Vcvv 2730  cun 3119  cin 3120   × cxp 4609  cres 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-res 4623
This theorem is referenced by:  imaundir  5024  fvunsng  5690  fvsnun1  5693  fvsnun2  5694  fsnunfv  5697  fsnunres  5698  fseq1p1m1  10050  setsresg  12454  setscom  12456  setsslid  12466
  Copyright terms: Public domain W3C validator