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Metamath Proof Explorer

Theorem neif

Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007) (Revised by Mario Carneiro, 11-Nov-2013)

Ref Expression
Hypothesis neifval.1 
|- X = U. J
Assertion neif 
|- ( J e. Top -> ( nei ` J ) Fn ~P X )

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 rabexg 
 |-  ( ~P X e. _V -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
5 2 3 4 3syl 
 |-  ( J e. Top -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
6 5 ralrimivw 
 |-  ( J e. Top -> A. x e. ~P X { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
7 eqid 
 |-  ( x e. ~P X |-> { v e. ~P X | 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 ) } )
8 7 fnmpt 
 |-  ( A. x e. ~P X { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V -> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X )
9 6 8 syl 
 |-  ( J e. Top -> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X )
10 1 neifval 
 |-  ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
11 10 fneq1d 
 |-  ( J e. Top -> ( ( nei ` J ) Fn ~P X <-> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X ) )
12 9 11 mpbird 
 |-  ( J e. Top -> ( nei ` J ) Fn ~P X )