Theorem opres

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Hypothesis	opres.1	⊢ 𝐵 ∈ V
Assertion	opres	⊢ ( 𝐴 ∈ 𝐷 → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 ↾ 𝐷 ) ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		opres.1	⊢ 𝐵 ∈ V
2	1	opelresi	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 ↾ 𝐷 ) ↔ ( 𝐴 ∈ 𝐷 ∧ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐶 ) )
3	2	baib	⊢ ( 𝐴 ∈ 𝐷 → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝐶 ↾ 𝐷 ) ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐶 ) )