Theorem lpss2

Description: Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Hypothesis	lpss2.1	⊢ 𝑋 = ∪ 𝐽
Assertion	lpss2	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝐵 ) ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lpss2.1	⊢ 𝑋 = ∪ 𝐽
2	1	lpss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( limPt ‘ 𝐽 ) ‘ 𝐵 ) ⊆ ( ( limPt ‘ 𝐽 ) ‘ 𝐴 ) )