Theorem relin2 5806

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 4224	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		relss 5774	. 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 3942   ⊆ wss 3943  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-rel 5676

This theorem is referenced by:  relinxp  5807  intasym  6109  asymref  6110  poirr2  6118  symgcom2  32749  cnvref4  37730  dfantisymrel4  38142  dfantisymrel5  38143  clcnvlem  42931