Theorem neifval 12610

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 12476	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4143	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5694	. . 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 3783	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2208	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3549	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		rexeq 2653	. . . . 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 2707	. . . 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 4048	. . 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 12609	. . 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 5546	. 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 418	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 103   = wceq 1335   e. wcel 2128   E.wrex 2436   {crab 2439   _Vcvv 2712   C_ wss 3102   ~Pcpw 3544   U.cuni 3774   |-> cmpt 4027   ` cfv 5172   Topctop 12465   neicnei 12608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-top 12466  df-nei 12609

This theorem is referenced by:  neif  12611  neival  12613