ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relin2 GIF version

Theorem relin2 4628
Description: The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
relin2 (Rel 𝐵 → Rel (𝐴𝐵))

Proof of Theorem relin2
StepHypRef Expression
1 inss2 3267 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 4596 . 2 ((𝐴𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐵 → Rel (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  cin 3040  wss 3041  Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-rel 4516
This theorem is referenced by:  intasym  4893  asymref  4894  poirr2  4901
  Copyright terms: Public domain W3C validator