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Metamath Proof Explorer

Theorem clsf

Description: The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015)

Ref Expression
Hypothesis clscld.1 
|- X = U. J
Assertion clsf 
|- ( J e. Top -> ( cls ` J ) : ~P X --> ( Clsd ` J ) )

Proof

Step Hyp Ref Expression
1 clscld.1 
 |-  X = U. J
2 elpwi 
 |-  ( x e. ~P X -> x C_ X )
3 1 clsval 
 |-  ( ( J e. Top /\ x C_ X ) -> ( ( cls ` J ) ` x ) = |^| { y e. ( Clsd ` J ) | x C_ y } )
4 fvex 
 |-  ( ( cls ` J ) ` x ) e. _V
5 3 4 eqeltrrdi 
 |-  ( ( J e. Top /\ x C_ X ) -> |^| { y e. ( Clsd ` J ) | x C_ y } e. _V )
6 2 5 sylan2 
 |-  ( ( J e. Top /\ x e. ~P X ) -> |^| { y e. ( Clsd ` J ) | x C_ y } e. _V )
7 1 clsfval 
 |-  ( J e. Top -> ( cls ` J ) = ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) )
8 elpwi 
 |-  ( y e. ~P X -> y C_ X )
9 1 clscld 
 |-  ( ( J e. Top /\ y C_ X ) -> ( ( cls ` J ) ` y ) e. ( Clsd ` J ) )
10 8 9 sylan2 
 |-  ( ( J e. Top /\ y e. ~P X ) -> ( ( cls ` J ) ` y ) e. ( Clsd ` J ) )
11 6 7 10 fmpt2d 
 |-  ( J e. Top -> ( cls ` J ) : ~P X --> ( Clsd ` J ) )