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Theorem relin2 4484
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 3186 . 2 (𝐴𝐵) ⊆ 𝐵
2 relss 4455 . 2 ((𝐴𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴𝐵)))
31, 2ax-mp 7 1 (Rel 𝐵 → Rel (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  cin 2944  wss 2945  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-rel 4380
This theorem is referenced by:  intasym  4737  asymref  4738  poirr2  4745
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