Theorem relin2 5723

Description: The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
relin2	⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))

Proof of Theorem relin2

Step	Hyp	Ref	Expression
1		inss2 4163	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		relss 5692	. 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → (Rel 𝐵 → Rel (𝐴 ∩ 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∩ cin 3886   ⊆ wss 3887  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-rel 5596

This theorem is referenced by:  relinxp  5724  intasym  6020  asymref  6021  poirr2  6029  symgcom2  31353  clcnvlem  41231