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Metamath Proof Explorer

Theorem clsfval

Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis cldval.1 
|- X = U. J
Assertion clsfval 
|- ( J e. Top -> ( cls ` J ) = ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) )

Proof

Step Hyp Ref Expression
1 cldval.1 
 |-  X = U. J
2 1 topopn 
 |-  ( J e. Top -> X e. J )
3 pwexg 
 |-  ( X e. J -> ~P X e. _V )
4 mptexg 
 |-  ( ~P X e. _V -> ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) e. _V )
5 2 3 4 3syl 
 |-  ( J e. Top -> ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) e. _V )
6 unieq 
 |-  ( j = J -> U. j = U. J )
7 6 1 eqtr4di 
 |-  ( j = J -> U. j = X )
8 7 pweqd 
 |-  ( j = J -> ~P U. j = ~P X )
9 fveq2 
 |-  ( j = J -> ( Clsd ` j ) = ( Clsd ` J ) )
10 9 rabeqdv 
 |-  ( j = J -> { y e. ( Clsd ` j ) | x C_ y } = { y e. ( Clsd ` J ) | x C_ y } )
11 10 inteqd 
 |-  ( j = J -> |^| { y e. ( Clsd ` j ) | x C_ y } = |^| { y e. ( Clsd ` J ) | x C_ y } )
12 8 11 mpteq12dv 
 |-  ( j = J -> ( x e. ~P U. j |-> |^| { y e. ( Clsd ` j ) | x C_ y } ) = ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) )
13 df-cls 
 |-  cls = ( j e. Top |-> ( x e. ~P U. j |-> |^| { y e. ( Clsd ` j ) | x C_ y } ) )
14 12 13 fvmptg 
 |-  ( ( J e. Top /\ ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) e. _V ) -> ( cls ` J ) = ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) )
15 5 14 mpdan 
 |-  ( J e. Top -> ( cls ` J ) = ( x e. ~P X |-> |^| { y e. ( Clsd ` J ) | x C_ y } ) )