Theorem relin2 5757

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 4187	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		relss 5726	. 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 3897   ⊆ wss 3898  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-in 3905  df-ss 3915  df-rel 5626

This theorem is referenced by:  relinxp  5758  intasym  6066  asymref  6067  poirr2  6075  symgcom2  33060  cnvref4  38403  dfantisymrel4  38880  dfantisymrel5  38881  clcnvlem  43741