Metamath Proof Explorer


Theorem dmoprab

Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995) (Revised by David Abernethy, 19-Jun-2012)

Ref Expression
Assertion dmoprab dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 𝜑 }

Proof

Step Hyp Ref Expression
1 dfoprab2 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
2 1 dmeqi dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = dom { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
3 dmopab dom { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑧𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
4 exrot3 ( ∃ 𝑧𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥𝑦𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
5 19.42v ( ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 𝜑 ) )
6 5 2exbii ( ∃ 𝑥𝑦𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 𝜑 ) )
7 4 6 bitri ( ∃ 𝑧𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 𝜑 ) )
8 7 abbii { 𝑤 ∣ ∃ 𝑧𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 𝜑 ) }
9 df-opab { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 𝜑 } = { 𝑤 ∣ ∃ 𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 𝜑 ) }
10 8 9 eqtr4i { 𝑤 ∣ ∃ 𝑧𝑥𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 𝜑 }
11 2 3 10 3eqtri dom { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 𝜑 }