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Metamath Proof Explorer

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 ) )