Theorem eliin

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

	Ref	Expression
Assertion	eliin	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1w	⊢ ( 𝑦 = 𝑤 → ( 𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐶 ) )
2	1	ralbidv	⊢ ( 𝑦 = 𝑤 → ( ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑤 ∈ 𝐶 ) )
3		eleq1	⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
4	3	ralbidv	⊢ ( 𝑤 = 𝐴 → ( ∀ 𝑥 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )
5		df-iin	⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = { 𝑦 ∣ ∀ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 }
6	2 4 5	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) )