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 }