Theorem cldlp

Description: A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of Munkres p. 97. (Contributed by NM, 26-Feb-2007)

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

Proof

Step	Hyp	Ref	Expression
1		lpfval.1	|- X = U. J
2	1	iscld3	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) = S ) )
3	1	clslp	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( S u. ( ( limPt ` J ) ` S ) ) )
4	3	eqeq1d	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) = S <-> ( S u. ( ( limPt ` J ) ` S ) ) = S ) )
5		ssequn2	|- ( ( ( limPt ` J ) ` S ) C_ S <-> ( S u. ( ( limPt ` J ) ` S ) ) = S )
6	4 5	bitr4di	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) = S <-> ( ( limPt ` J ) ` S ) C_ S ) )
7	2 6	bitrd	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( limPt ` J ) ` S ) C_ S ) )