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

Theorem resindir 4938
Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
resindir ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem resindir
StepHypRef Expression
1 inindir 3368 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 4653 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4653 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4653 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4ineq12i 3349 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2220 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1364  Vcvv 2752  cin 3143   × cxp 4639  cres 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-res 4653
This theorem is referenced by:  inimass  5060  fnreseql  5642
  Copyright terms: Public domain W3C validator