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Metamath Proof Explorer

Theorem opreu2reurex

Description: There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023) (Revised by AV, 1-Jul-2023)

Ref Expression
Hypothesis opreu2reurex.a ⊢ ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝜑 ↔ 𝜒 ) )
Assertion opreu2reurex ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) )

Proof

Step Hyp Ref Expression
1 opreu2reurex.a ⊢ ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ → ( 𝜑 ↔ 𝜒 ) )
2 eqcom ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ↔ ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
3 vex ⊢ 𝑎 ∈ V
4 vex ⊢ 𝑏 ∈ V
5 3 4 opth ⊢ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) )
6 2 5 bitri ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ↔ ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) )
7 6 imbi2i ⊢ ( ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) )
8 7 a1i ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ) ) → ( ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
9 8 2ralbidva ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
10 9 2rexbiia ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) )
11 10 anbi2i ⊢ ( ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ) ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
12 1 reu3op ⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ) )
13 2reu4 ⊢ ( ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) ↔ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∀ 𝑎 ∈ 𝐴 ∀ 𝑏 ∈ 𝐵 ( 𝜒 → ( 𝑎 = 𝑥 ∧ 𝑏 = 𝑦 ) ) ) )
14 11 12 13 3bitr4i ⊢ ( ∃! 𝑝 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ( ∃! 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 𝜒 ∧ ∃! 𝑏 ∈ 𝐵 ∃ 𝑎 ∈ 𝐴 𝜒 ) )