Theorem relin2 5823

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 4238	. 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
2		relss 5791	. 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 3950   ⊆ wss 3951  Rel wrel 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-rel 5692

This theorem is referenced by:  relinxp  5824  intasym  6135  asymref  6136  poirr2  6144  symgcom2  33104  cnvref4  38351  dfantisymrel4  38762  dfantisymrel5  38763  clcnvlem  43636