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Metamath Proof Explorer

Theorem iscld4

Description: A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008)

Ref Expression
Hypothesis clscld.1 
|- X = U. J
Assertion iscld4 
|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) C_ S ) )

Proof

Step Hyp Ref Expression
1 clscld.1 
 |-  X = U. J
2 1 iscld3 
 |-  ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) = S ) )
3 eqss 
 |-  ( ( ( cls ` J ) ` S ) = S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) )
4 1 sscls 
 |-  ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
5 4 biantrud 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) C_ S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) ) )
6 3 5 bitr4id 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) = S <-> ( ( cls ` J ) ` S ) C_ S ) )
7 2 6 bitrd 
 |-  ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) C_ S ) )