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 = U. J
Assertion	lpsscls	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ ( ( cls ` J ) ` S ) )

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	lpval	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
3		difss	|- ( S \ { x } ) C_ S
4	1	clsss	|- ( ( J e. Top /\ S C_ X /\ ( S \ { x } ) C_ S ) -> ( ( cls ` J ) ` ( S \ { x } ) ) C_ ( ( cls ` J ) ` S ) )
5	3 4	mp3an3	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( S \ { x } ) ) C_ ( ( cls ` J ) ` S ) )
6	5	sseld	|- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) -> x e. ( ( cls ` J ) ` S ) ) )
7	6	abssdv	|- ( ( J e. Top /\ S C_ X ) -> { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } C_ ( ( cls ` J ) ` S ) )
8	2 7	eqsstrd	|- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) C_ ( ( cls ` J ) ` S ) )