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 1 sscls
 |-  ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
4 3 biantrud
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) C_ S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) ) )
5 eqss
 |-  ( ( ( cls ` J ) ` S ) = S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) )
6 4 5 syl6rbbr
 |-  ( ( 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 ) )