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Theorem relin2 5767
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 4187 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 5735 . 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 3907  wss 3908  Rel wrel 5636
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-rel 5638
This theorem is referenced by:  relinxp  5768  intasym  6067  asymref  6068  poirr2  6076  symgcom2  31777  cnvref4  36743  dfantisymrel4  37155  dfantisymrel5  37156  clcnvlem  41800
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