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Metamath Proof Explorer

Theorem clsidm

Description: The closure operation is idempotent. (Contributed by NM, 2-Oct-2007)

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

Proof

Step Hyp Ref Expression
1 clscld.1 
 |-  X = U. J
2 1 clscld 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) e. ( Clsd ` J ) )
3 1 clsss3 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4 1 iscld3 
 |-  ( ( J e. Top /\ ( ( cls ` J ) ` S ) C_ X ) -> ( ( ( cls ` J ) ` S ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) ) )
5 3 4 syldan 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) ) )
6 2 5 mpbid 
 |-  ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) )