Theorem neival 14322

Description: Value of the set of neighborhoods of a subset 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
neival	|- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )

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

Proof of Theorem neival

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	neifval 14319	. . . 4 |- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
3	2	fveq1d 5557	. . 3 |- ( J e. Top -> ( ( nei ` J ) ` S ) = ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) )
4	3	adantr 276	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) )
5	1	topopn 14187	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4186	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 14	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 297	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		pwexg 4210	. . . . 5 |- ( X e. J -> ~P X e. _V )
10		rabexg 4173	. . . . 5 |- ( ~P X e. _V -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
11	5, 9, 10	3syl 17	. . . 4 |- ( J e. Top -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
12	11	adantr 276	. . 3 |- ( ( J e. Top /\ S C_ X ) -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
13		sseq1 3203	. . . . . . 7 |- ( x = S -> ( x C_ g <-> S C_ g ) )
14	13	anbi1d 465	. . . . . 6 |- ( x = S -> ( ( x C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ v ) ) )
15	14	rexbidv 2495	. . . . 5 |- ( x = S -> ( E. g e. J ( x C_ g /\ g C_ v ) <-> E. g e. J ( S C_ g /\ g C_ v ) ) )
16	15	rabbidv 2749	. . . 4 |- ( x = S -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
17		eqid 2193	. . . 4 |- ( 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 ) } )
18	16, 17	fvmptg 5634	. . 3 |- ( ( S e. ~P X /\ { v e. ~P X | E. g e. J ( S 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 ) } ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
19	8, 12, 18	syl2anc 411	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
20	4, 19	eqtrd 2226	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = 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:  isnei  14323