Theorem relin1

Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	relin1	⊢ ( Rel 𝐴 → Rel ( 𝐴 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2		relss	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 → ( Rel 𝐴 → Rel ( 𝐴 ∩ 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( Rel 𝐴 → Rel ( 𝐴 ∩ 𝐵 ) )