Theorem eliun

Description: Membership in indexed union. (Contributed by NM, 3-Sep-2003)

	Ref	Expression
Assertion	eliun	⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 → 𝐴 ∈ V )
2		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
3	2	rexlimivw	⊢ ( ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
4		eleq1	⊢ ( 𝑦 = 𝑤 → ( 𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐶 ) )
5	4	rexbidv	⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑤 ∈ 𝐶 ) )
6		eleq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
7	6	rexbidv	⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑥 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )
8		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 }
9	5 7 8	elab2gw	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )
10	1 3 9	pm5.21nii	⊢ ( 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 )