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Metamath Proof Explorer

Theorem opabid2

Description: A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006)

Ref Expression
Assertion opabid2 ( Rel 𝐴 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } = 𝐴 )

Proof

Step Hyp Ref Expression
1 vex ⊢ 𝑧 ∈ V
2 vex ⊢ 𝑤 ∈ V
3 opeq1 ⊢ ( 𝑥 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑦 ⟩ )
4 3 eleq1d ⊢ ( 𝑥 = 𝑧 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ) )
5 opeq2 ⊢ ( 𝑦 = 𝑤 → ⟨ 𝑧 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ )
6 5 eleq1d ⊢ ( 𝑦 = 𝑤 → ( ⟨ 𝑧 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ 𝐴 ) )
7 1 2 4 6 opelopab ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ 𝐴 )
8 7 gen2 ⊢ ∀ 𝑧 ∀ 𝑤 ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ 𝐴 )
9 relopabv ⊢ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 }
10 eqrel ⊢ ( ( Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } ∧ Rel 𝐴 ) → ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } = 𝐴 ↔ ∀ 𝑧 ∀ 𝑤 ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ 𝐴 ) ) )
11 9 10 mpan ⊢ ( Rel 𝐴 → ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } = 𝐴 ↔ ∀ 𝑧 ∀ 𝑤 ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ 𝐴 ) ) )
12 8 11 mpbiri ⊢ ( Rel 𝐴 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 } = 𝐴 )