Theorem fclssscls

Description: The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015) (Revised by Stefan O'Rear, 8-Aug-2015)

	Ref	Expression
Assertion	fclssscls	⊢ ( 𝑆 ∈ 𝐹 → ( 𝐽 fClus 𝐹 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isfcls	⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐽 ∈ Top ∧ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ∧ ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
3	2	simp3bi	⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) )
4		fveq2	⊢ ( 𝑠 = 𝑆 → ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) = ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )
5	4	eleq2d	⊢ ( 𝑠 = 𝑆 → ( 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
6	5	rspcv	⊢ ( 𝑆 ∈ 𝐹 → ( ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
7	3 6	syl5	⊢ ( 𝑆 ∈ 𝐹 → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
8	7	ssrdv	⊢ ( 𝑆 ∈ 𝐹 → ( 𝐽 fClus 𝐹 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )