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Metamath Proof Explorer

Theorem unopab

Description: Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002)

Ref Expression
Assertion unopab ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∨ 𝜓 ) }

Proof

Step Hyp Ref Expression
1 eqeq1 ⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ) )
2 1 anbi1d ⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
3 2 2exbidv ⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
4 1 anbi1d ⊢ ( 𝑧 = 𝑤 → ( ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
5 4 2exbidv ⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
6 3 5 unabw ⊢ ( { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } ∪ { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } ) = { 𝑤 ∣ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) }
7 19.43 ⊢ ( ∃ 𝑥 ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
8 andi ⊢ ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) ↔ ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
9 8 exbii ⊢ ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) ↔ ∃ 𝑦 ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
10 19.43 ⊢ ( ∃ 𝑦 ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) ↔ ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
11 9 10 bitr2i ⊢ ( ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) )
12 11 exbii ⊢ ( ∃ 𝑥 ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) )
13 7 12 bitr3i ⊢ ( ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) )
14 13 abbii ⊢ { 𝑤 ∣ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ∨ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) }
15 6 14 eqtri ⊢ ( { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } ∪ { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } ) = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) }
16 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
17 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) }
18 16 17 uneq12i ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) = ( { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } ∪ { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } )
19 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∨ 𝜓 ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝜑 ∨ 𝜓 ) ) }
20 15 18 19 3eqtr4i ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝜑 ∨ 𝜓 ) }