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Theorem relin2 5659
 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 4136 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 5629 . 2 ((𝐴𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐵 → Rel (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∩ cin 3859   ⊆ wss 3860  Rel wrel 5532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-in 3867  df-ss 3877  df-rel 5534 This theorem is referenced by:  relinxp  5660  intasym  5951  asymref  5952  poirr2  5960  symgcom2  30883  clcnvlem  40724
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