Theorem brresi

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

	Ref	Expression
Hypothesis	opelresi.1	⊢ 𝐶 ∈ V
Assertion	brresi	⊢ ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		opelresi.1	⊢ 𝐶 ∈ V
2		brres	⊢ ( 𝐶 ∈ V → ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )
3	1 2	ax-mp	⊢ ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) )