Theorem brinxprnres

Description: Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019)

	Ref	Expression
Assertion	brinxprnres	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brres2	⊢ ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ 𝐵 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝐶 )
2		brres	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )
3	1 2	bitr3id	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )