Theorem relres

Description: A restriction is a relation. Exercise 12 of TakeutiZaring p. 25. (Contributed by NM, 2-Aug-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Proof shortened by BJ, 16-Oct-2026)

	Ref	Expression
Assertion	relres	⊢ Rel ( 𝐴 ↾ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		relinxp	⊢ Rel ( 𝐴 ∩ ( 𝐵 × V ) )
2		df-res	⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) )
3	2	releqi	⊢ ( Rel ( 𝐴 ↾ 𝐵 ) ↔ Rel ( 𝐴 ∩ ( 𝐵 × V ) ) )
4	1 3	mpbir	⊢ Rel ( 𝐴 ↾ 𝐵 )