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Metamath Proof Explorer

Theorem neifval

Description: Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis neifval.1 
|- X = U. J
Assertion neifval 
|- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )

Proof

Step Hyp Ref Expression
1 neifval.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 |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) e. _V )
5 2 3 4 3syl 
 |-  ( J e. Top -> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) 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 rexeq 
 |-  ( j = J -> ( E. g e. j ( x C_ g /\ g C_ v ) <-> E. g e. J ( x C_ g /\ g C_ v ) ) )
10 8 9 rabeqbidv 
 |-  ( j = J -> { v e. ~P U. j | E. g e. j ( x C_ g /\ g C_ v ) } = { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } )
11 8 10 mpteq12dv 
 |-  ( j = J -> ( x e. ~P U. j |-> { v e. ~P U. j | E. g e. j ( x C_ g /\ g C_ v ) } ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
12 df-nei 
 |-  nei = ( j e. Top |-> ( x e. ~P U. j |-> { v e. ~P U. j | E. g e. j ( x C_ g /\ g C_ v ) } ) )
13 11 12 fvmptg 
 |-  ( ( J e. Top /\ ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) e. _V ) -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
14 5 13 mpdan 
 |-  ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )