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Metamath Proof Explorer

Theorem rnoprab

Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004) (Revised by David Abernethy, 19-Apr-2013)

Ref Expression
Assertion rnoprab ran { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 𝜑 }

Proof

Step Hyp Ref Expression
1 dfoprab2 ⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
2 1 rneqi ⊢ ran { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = ran { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
3 rnopab ⊢ ran { ⟨ 𝑤 , 𝑧 ⟩ ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑧 ∣ ∃ 𝑤 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
4 exrot3 ⊢ ( ∃ 𝑤 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑤 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
5 opex ⊢ ⟨ 𝑥 , 𝑦 ⟩ ∈ V
6 5 isseti ⊢ ∃ 𝑤 𝑤 = ⟨ 𝑥 , 𝑦 ⟩
7 19.41v ⊢ ( ∃ 𝑤 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ∃ 𝑤 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
8 6 7 mpbiran ⊢ ( ∃ 𝑤 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜑 )
9 8 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑤 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 𝜑 )
10 4 9 bitri ⊢ ( ∃ 𝑤 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 𝜑 )
11 10 abbii ⊢ { 𝑧 ∣ ∃ 𝑤 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 𝜑 }
12 2 3 11 3eqtri ⊢ ran { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 𝜑 }