Theorem dfoprab4

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	dfoprab4.1	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ 𝜓 ) )
Assertion	dfoprab4	⊢ { ⟨ 𝑤 , 𝑧 ⟩ ∣ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) }

Proof

Step	Hyp	Ref	Expression
1		dfoprab4.1	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ 𝜓 ) )
2		xpss	⊢ ( 𝐴 × 𝐵 ) ⊆ ( V × V )
3	2	sseli	⊢ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) → 𝑤 ∈ ( V × V ) )
4	3	adantr	⊢ ( ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) → 𝑤 ∈ ( V × V ) )
5	4	pm4.71ri	⊢ ( ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑤 ∈ ( V × V ) ∧ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ) )
6	5	opabbii	⊢ { ⟨ 𝑤 , 𝑧 ⟩ ∣ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } = { ⟨ 𝑤 , 𝑧 ⟩ ∣ ( 𝑤 ∈ ( V × V ) ∧ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ) }
7		eleq1	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ) )
8		opelxp	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) )
9	7 8	syl6bb	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) )
10	9 1	anbi12d	⊢ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ → ( ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) ) )
11	10	dfoprab3	⊢ { ⟨ 𝑤 , 𝑧 ⟩ ∣ ( 𝑤 ∈ ( V × V ) ∧ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) }
12	6 11	eqtri	⊢ { ⟨ 𝑤 , 𝑧 ⟩ ∣ ( 𝑤 ∈ ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜓 ) }