Theorem neifval 14697

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 14565	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4235	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5827	. . 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 3868	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2257	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3626	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		rexeq 2704	. . . . 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 2768	. . . 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 4137	. . 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 14696	. . 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 5673	. 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 1373   e. wcel 2177   E.wrex 2486   {crab 2489   _Vcvv 2773   C_ wss 3170   ~Pcpw 3621   U.cuni 3859   |-> cmpt 4116   ` cfv 5285   Topctop 14554   neicnei 14695

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-top 14555  df-nei 14696

This theorem is referenced by:  neif  14698  neival  14700