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Theorem relin2 5815
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 4230 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 5783 . 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 3946  wss 3947  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-rel 5685
This theorem is referenced by:  relinxp  5816  intasym  6121  asymref  6122  poirr2  6130  symgcom2  32820  cnvref4  37822  dfantisymrel4  38233  dfantisymrel5  38234  clcnvlem  43053
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