Theorem lpsscls

Description: The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007)

	Ref	Expression
Hypothesis	lpfval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	lpsscls	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	⊢ 𝑋 = ∪ 𝐽
2	1	lpval	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) = { 𝑥 ∣ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) } )
3		difss	⊢ ( 𝑆 ∖ { 𝑥 } ) ⊆ 𝑆
4	1	clsss	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ ( 𝑆 ∖ { 𝑥 } ) ⊆ 𝑆 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )
5	3 4	mp3an3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )
6	5	sseld	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
7	6	abssdv	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → { 𝑥 ∣ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ ( 𝑆 ∖ { 𝑥 } ) ) } ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )
8	2 7	eqsstrd	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )