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	$$⊢\mathrm{Rel}\left({A\upharpoonright }_B\right)$$

Proof

Step	Hyp	Ref	Expression
1		relinxp	$$⊢\mathrm{Rel}\left(A\cap \left(B\times \mathrm{V}\right)\right)$$
2		df-res	$$⊢{A\upharpoonright }_B=A\cap \left(B\times \mathrm{V}\right)$$
3	2	releqi	$$⊢\mathrm{Rel}\left({A\upharpoonright }_B\right)\leftrightarrow \mathrm{Rel}\left(A\cap \left(B\times \mathrm{V}\right)\right)$$
4	1 3	mpbir	$$⊢\mathrm{Rel}\left({A\upharpoonright }_B\right)$$