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Theorem relin1 5825
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 4245 . 2 (𝐴𝐵) ⊆ 𝐴
2 relss 5794 . 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 3962  wss 3963  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-ss 3980  df-rel 5696
This theorem is referenced by:  inopab  5842  idsset  35872  dihmeetlem1N  41273  dihglblem5apreN  41274  dihmeetlem4preN  41289  dihmeetlem13N  41302
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