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Theorem relin1 5159
 Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
relin1 (Rel 𝐴 → Rel (𝐴𝐵))

Proof of Theorem relin1
StepHypRef Expression
1 inss1 3795 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5129 . 2 ((𝐴𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐴 → Rel (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3539   ⊆ wss 3540  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-rel 5045 This theorem is referenced by:  inopab  5174  idsset  31167  dihmeetlem1N  35597  dihglblem5apreN  35598  dihmeetlem4preN  35613  dihmeetlem13N  35626
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