Theorem neival 14817

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 14814	. . . 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 5629	. . 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 14682	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4240	. . . . 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 4264	. . . . 5 |- ( X e. J -> ~P X e. _V )
10		rabexg 4227	. . . . 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 3247	. . . . . . 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 2531	. . . . 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 2788	. . . 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 2229	. . . 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 5710	. . 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 2262	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 1395   e. wcel 2200   E.wrex 2509   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   U.cuni 3888   |-> cmpt 4145   ` cfv 5318   Topctop 14671   neicnei 14812

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-top 14672  df-nei 14813

This theorem is referenced by:  isnei  14818