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Theorem relin2 4763
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 3371 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 4731 . 2 ((𝐴𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴𝐵)))
31, 2ax-mp 5 1 (Rel 𝐵 → Rel (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  cin 3143  wss 3144  Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-rel 4651
This theorem is referenced by:  intasym  5031  asymref  5032  poirr2  5039
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