Theorem neival 12793

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 12790	. . . 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 5488	. . 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 274	. 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 12656	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4135	. . . . 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 295	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		pwexg 4159	. . . . 5 |- ( X e. J -> ~P X e. _V )
10		rabexg 4125	. . . . 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 274	. . 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 3165	. . . . . . 7 |- ( x = S -> ( x C_ g <-> S C_ g ) )
14	13	anbi1d 461	. . . . . 6 |- ( x = S -> ( ( x C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ v ) ) )
15	14	rexbidv 2467	. . . . 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 2715	. . . 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 2165	. . . 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 5562	. . 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 409	. 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 2198	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   {crab 2448   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   |-> cmpt 4043   ` cfv 5188   Topctop 12645   neicnei 12788

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12646  df-nei 12789

This theorem is referenced by:  isnei  12794