Theorem neifval 14319

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.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

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

Distinct variable groups:   v, g, x, J   g, X, v, x

Proof of Theorem neifval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . 4 |- X = U. J
2	1	topopn 14187	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4210	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5784	. . 3 |- ( ~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 17	. 2 |- ( 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 3845	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2244	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3607	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		rexeq 2691	. . . . 5 |- ( 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 2755	. . . 4 |- ( 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 4112	. . 3 |- ( 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 14318	. . 3 |- 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 5634	. 2 |- ( ( 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 421	1 |- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   E.wrex 2473   {crab 2476   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   U.cuni 3836   |-> cmpt 4091   ` cfv 5255   Topctop 14176   neicnei 14317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 14177  df-nei 14318

This theorem is referenced by:  neif  14320  neival  14322