Theorem oprab4

Description: Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010)

	Ref	Expression
Assertion	oprab4	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) }

Proof

Step	Hyp	Ref	Expression
1		opelxp	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) )
2	1	anbi1i	⊢ ( ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) )
3	2	oprabbii	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) }