Theorem lpss

Description: The limit points of a subset are included in the base set. (Contributed by NM, 9-Nov-2007)

	Ref	Expression
Hypothesis	lpfval.1	|- X = U. J
Assertion	lpss	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ X )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	lpsscls	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ ( ( cls ` J ) ` S ) )
3	1	clsss3	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4	2 3	sstrd	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ X )