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 )