Theorem neifval 14376

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 14244	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4213	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5787	. . 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 3848	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2247	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3610	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		rexeq 2694	. . . . 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 2758	. . . 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 4115	. . 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 14375	. . 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 5637	. 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 2167   E.wrex 2476   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   U.cuni 3839   |-> cmpt 4094   ` cfv 5258   Topctop 14233   neicnei 14374

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-top 14234  df-nei 14375

This theorem is referenced by:  neif  14377  neival  14379