This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem iunopab

Description: Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015) Avoid ax-sep , ax-nul , ax-pr . (Revised by SN, 11-Nov-2024)

Ref Expression
Assertion iunopab ∪ 𝑧 ∈ 𝐴 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 ∈ 𝐴 𝜑 }

Proof

Step Hyp Ref Expression
1 elopabw ⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
2 1 elv ⊢ ( 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
3 2 rexbii ⊢ ( ∃ 𝑧 ∈ 𝐴 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
4 rexcom4 ⊢ ( ∃ 𝑧 ∈ 𝐴 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
5 rexcom4 ⊢ ( ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
6 r19.42v ⊢ ( ∃ 𝑧 ∈ 𝐴 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
7 6 exbii ⊢ ( ∃ 𝑦 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
8 5 7 bitri ⊢ ( ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
9 8 exbii ⊢ ( ∃ 𝑥 ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
10 4 9 bitri ⊢ ( ∃ 𝑧 ∈ 𝐴 ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
11 3 10 bitri ⊢ ( ∃ 𝑧 ∈ 𝐴 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) )
12 11 abbii ⊢ { 𝑤 ∣ ∃ 𝑧 ∈ 𝐴 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) }
13 df-iun ⊢ ∪ 𝑧 ∈ 𝐴 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑧 ∈ 𝐴 𝑤 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } }
14 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 ∈ 𝐴 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ∃ 𝑧 ∈ 𝐴 𝜑 ) }
15 12 13 14 3eqtr4i ⊢ ∪ 𝑧 ∈ 𝐴 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑧 ∈ 𝐴 𝜑 }