Metamath Proof Explorer


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 ( 𝑆𝐹 → ( 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 ) ∧ ∀ 𝑠𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
3 2 simp3bi ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → ∀ 𝑠𝐹 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 ( 𝑆𝐹 → ( ∀ 𝑠𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
7 3 6 syl5 ( 𝑆𝐹 → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
8 7 ssrdv ( 𝑆𝐹 → ( 𝐽 fClus 𝐹 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )