Theorem brresi

Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006)

	Ref	Expression
Hypothesis	opelresi.1	$$⊢C\in \mathrm{V}$$
Assertion	brresi	$$⊢B\left({R\upharpoonright }_A\right)C\leftrightarrow \left(B\in A\wedge BRC\right)$$

Proof

Step	Hyp	Ref	Expression
1		opelresi.1	$$⊢C\in \mathrm{V}$$
2		brres	$$⊢C\in \mathrm{V}\to \left(B\left({R\upharpoonright }_A\right)C\leftrightarrow \left(B\in A\wedge BRC\right)\right)$$
3	1 2	ax-mp	$$⊢B\left({R\upharpoonright }_A\right)C\leftrightarrow \left(B\in A\wedge BRC\right)$$