Theorem opelresi

Description: Ordered pair membership in a restriction. Exercise 13 of TakeutiZaring p. 25. (Contributed by NM, 13-Nov-1995)

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

Proof

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