Theorem relin2

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

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

Proof

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