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)

	Ref	Expression
Assertion	relres	⊢ Rel ( 𝐴 ↾ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-res	⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) )
2		inss2	⊢ ( 𝐴 ∩ ( 𝐵 × V ) ) ⊆ ( 𝐵 × V )
3	1 2	eqsstri	⊢ ( 𝐴 ↾ 𝐵 ) ⊆ ( 𝐵 × V )
4		relxp	⊢ Rel ( 𝐵 × V )
5		relss	⊢ ( ( 𝐴 ↾ 𝐵 ) ⊆ ( 𝐵 × V ) → ( Rel ( 𝐵 × V ) → Rel ( 𝐴 ↾ 𝐵 ) ) )
6	3 4 5	mp2	⊢ Rel ( 𝐴 ↾ 𝐵 )