Theorem neifg 15559

Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 15557.

Hypothesis

Ref	Expression
neifg.1	|- X = U.J

Assertion

Ref	Expression
neifg	|- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (filGen` {x e. J | S C_ x}) = ((nei` J)` S))

Distinct variable groups:   x,J   x,S   x,X

Proof of Theorem neifg

Step	Hyp	Ref	Expression
1		neifg.1	. . . 4 |- X = U.J
2	1	opnfbas 15557	. . 3 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> {x e. J | S C_ x} e. fBas)
3		eqid 1884	. . . 4 |- U.{x e. J | S C_ x} = U.{x e. J | S C_ x}
4	3	fgf 10283	. . 3 |- ({x e. J | S C_ x} e. fBas -> (filGen` {x e. J | S C_ x}) = {t e. ~PU.{x e. J | S C_ x} | ({x e. J | S C_ x} i^i ~Pt) =/= (/)})
5	2, 4	syl 12	. 2 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (filGen` {x e. J | S C_ x}) = {t e. ~PU.{x e. J | S C_ x} | ({x e. J | S C_ x} i^i ~Pt) =/= (/)})
6	1	eltopss 8872	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ y e. J) -> y C_ X)
7	6	ex 402	. . . . . . . . . . . . . 14 |- (J e. Top -> (y e. J -> y C_ X))
8	7	adantrd 427	. . . . . . . . . . . . 13 |- (J e. Top -> ((y e. J /\ S C_ y) -> y C_ X))
9	8	3ad2ant1 897	. . . . . . . . . . . 12 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> ((y e. J /\ S C_ y) -> y C_ X))
10		sseq2 2639	. . . . . . . . . . . . 13 |- (x = y -> (S C_ x <-> S C_ y))
11	10	elrab 2414	. . . . . . . . . . . 12 |- (y e. {x e. J | S C_ x} <-> (y e. J /\ S C_ y))
12	9, 11	syl5ib 223	. . . . . . . . . . 11 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (y e. {x e. J | S C_ x} -> y C_ X))
13	12	r19.21aiv 2175	. . . . . . . . . 10 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> A.y e. {x e. J | S C_ x}y C_ X)
14		unissb 3208	. . . . . . . . . 10 |- (U.{x e. J | S C_ x} C_ X <-> A.y e. {x e. J | S C_ x}y C_ X)
15	13, 14	sylibr 217	. . . . . . . . 9 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> U.{x e. J | S C_ x} C_ X)
16		sseq2 2639	. . . . . . . . . . . 12 |- (x = X -> (S C_ x <-> S C_ X))
17	16	elrab 2414	. . . . . . . . . . 11 |- (X e. {x e. J | S C_ x} <-> (X e. J /\ S C_ X))
18	1	topopn 8871	. . . . . . . . . . . 12 |- (J e. Top -> X e. J)
19	18	3ad2ant1 897	. . . . . . . . . . 11 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> X e. J)
20		simp2 877	. . . . . . . . . . 11 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> S C_ X)
21	17, 19, 20	sylanbrc 527	. . . . . . . . . 10 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> X e. {x e. J | S C_ x})
22		elssuni 3206	. . . . . . . . . 10 |- (X e. {x e. J | S C_ x} -> X C_ U.{x e. J | S C_ x})
23	21, 22	syl 12	. . . . . . . . 9 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> X C_ U.{x e. J | S C_ x})
24	15, 23	eqssd 2633	. . . . . . . 8 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> U.{x e. J | S C_ x} = X)
25	24	sseq2d 2645	. . . . . . 7 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (u C_ U.{x e. J | S C_ x} <-> u C_ X))
26		visset 2295	. . . . . . . 8 |- u e. _V
27	26	elpw 3037	. . . . . . 7 |- (u e. ~PU.{x e. J | S C_ x} <-> u C_ U.{x e. J | S C_ x})
28	25, 27	syl5bb 591	. . . . . 6 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (u e. ~PU.{x e. J | S C_ x} <-> u C_ X))
29		n0 2884	. . . . . . . 8 |- (({x e. J | S C_ x} i^i ~Pu) =/= (/) <-> E.z z e. ({x e. J | S C_ x} i^i ~Pu))
30		elin 2786	. . . . . . . . . 10 |- (z e. ({x e. J | S C_ x} i^i ~Pu) <-> (z e. {x e. J | S C_ x} /\ z e. ~Pu))
31		sseq2 2639	. . . . . . . . . . . 12 |- (x = z -> (S C_ x <-> S C_ z))
32	31	elrab 2414	. . . . . . . . . . 11 |- (z e. {x e. J | S C_ x} <-> (z e. J /\ S C_ z))
33		visset 2295	. . . . . . . . . . . 12 |- z e. _V
34	33	elpw 3037	. . . . . . . . . . 11 |- (z e. ~Pu <-> z C_ u)
35	32, 34	anbi12i 540	. . . . . . . . . 10 |- ((z e. {x e. J | S C_ x} /\ z e. ~Pu) <-> ((z e. J /\ S C_ z) /\ z C_ u))
36	30, 35	bitri 190	. . . . . . . . 9 |- (z e. ({x e. J | S C_ x} i^i ~Pu) <-> ((z e. J /\ S C_ z) /\ z C_ u))
37	36	exbii 1398	. . . . . . . 8 |- (E.z z e. ({x e. J | S C_ x} i^i ~Pu) <-> E.z((z e. J /\ S C_ z) /\ z C_ u))
38	29, 37	bitri 190	. . . . . . 7 |- (({x e. J | S C_ x} i^i ~Pu) =/= (/) <-> E.z((z e. J /\ S C_ z) /\ z C_ u))
39	38	a1i 8	. . . . . 6 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (({x e. J | S C_ x} i^i ~Pu) =/= (/) <-> E.z((z e. J /\ S C_ z) /\ z C_ u)))
40	28, 39	anbi12d 690	. . . . 5 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> ((u e. ~PU.{x e. J | S C_ x} /\ ({x e. J | S C_ x} i^i ~Pu) =/= (/)) <-> (u C_ X /\ E.z((z e. J /\ S C_ z) /\ z C_ u))))
41	1	isnei 8994	. . . . . . 7 |- ((J e. Top /\ S C_ X) -> (u e. ((nei` J)` S) <-> (u C_ X /\ E.z e. J (S C_ z /\ z C_ u))))
42	41	3adant3 896	. . . . . 6 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (u e. ((nei` J)` S) <-> (u C_ X /\ E.z e. J (S C_ z /\ z C_ u))))
43		anass 487	. . . . . . . . 9 |- (((z e. J /\ S C_ z) /\ z C_ u) <-> (z e. J /\ (S C_ z /\ z C_ u)))
44	43	exbii 1398	. . . . . . . 8 |- (E.z((z e. J /\ S C_ z) /\ z C_ u) <-> E.z(z e. J /\ (S C_ z /\ z C_ u)))
45		df-rex 2110	. . . . . . . 8 |- (E.z e. J (S C_ z /\ z C_ u) <-> E.z(z e. J /\ (S C_ z /\ z C_ u)))
46	44, 45	bitr4i 193	. . . . . . 7 |- (E.z((z e. J /\ S C_ z) /\ z C_ u) <-> E.z e. J (S C_ z /\ z C_ u))
47	46	anbi2i 538	. . . . . 6 |- ((u C_ X /\ E.z((z e. J /\ S C_ z) /\ z C_ u)) <-> (u C_ X /\ E.z e. J (S C_ z /\ z C_ u)))
48	42, 47	syl6rbbr 598	. . . . 5 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> ((u C_ X /\ E.z((z e. J /\ S C_ z) /\ z C_ u)) <-> u e. ((nei` J)` S)))
49	40, 48	bitrd 587	. . . 4 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> ((u e. ~PU.{x e. J | S C_ x} /\ ({x e. J | S C_ x} i^i ~Pu) =/= (/)) <-> u e. ((nei` J)` S)))
50		pweq 3036	. . . . . . 7 |- (t = u -> ~Pt = ~Pu)
51	50	ineq2d 2796	. . . . . 6 |- (t = u -> ({x e. J | S C_ x} i^i ~Pt) = ({x e. J | S C_ x} i^i ~Pu))
52	51	neeq1d 2028	. . . . 5 |- (t = u -> (({x e. J | S C_ x} i^i ~Pt) =/= (/) <-> ({x e. J | S C_ x} i^i ~Pu) =/= (/)))
53	52	elrab 2414	. . . 4 |- (u e. {t e. ~PU.{x e. J | S C_ x} | ({x e. J | S C_ x} i^i ~Pt) =/= (/)} <-> (u e. ~PU.{x e. J | S C_ x} /\ ({x e. J | S C_ x} i^i ~Pu) =/= (/)))
54	49, 53	syl5bb 591	. . 3 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (u e. {t e. ~PU.{x e. J | S C_ x} | ({x e. J | S C_ x} i^i ~Pt) =/= (/)} <-> u e. ((nei` J)` S)))
55	54	eqrdv 1882	. 2 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> {t e. ~PU.{x e. J | S C_ x} | ({x e. J | S C_ x} i^i ~Pt) =/= (/)} = ((nei` J)` S))
56	5, 55	eqtrd 1925	1 |- ((J e. Top /\ S C_ X /\ S =/= (/)) -> (filGen` {x e. J | S C_ x}) = ((nei` J)` S))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   i^i cin 2592   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988  fBascfbas 10257  filGencfg 10258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-nei 8989  df-fbas 10259  df-fg 10260

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