Theorem iunexg

Description: The existence of an indexed union. x is normally a free-variable parameter in B . (Contributed by NM, 23-Mar-2006)

	Ref	Expression
Assertion	iunexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		dfiun2g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } )
2	1	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } )
3		abrexexg	⊢ ( 𝐴 ∈ 𝑉 → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V )
4	3	uniexd	⊢ ( 𝐴 ∈ 𝑉 → ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V )
5	4	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ) → ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V )
6	2 5	eqeltrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V )