Theorem neiplim 15586

Description: A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.)

Hypothesis

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

Assertion

Ref	Expression
neiplim	|- ((J e. Top /\ A e. X) -> A e. ((fLim1` J)` ((nei` J)` {A})))

Proof of Theorem neiplim

Step	Hyp	Ref	Expression
1		ssid 2634	. . . 4 |- ((nei` J)` {A}) C_ ((nei` J)` {A})
2	1	jctr 315	. . 3 |- (A e. X -> (A e. X /\ ((nei` J)` {A}) C_ ((nei` J)` {A})))
3	2	adantl 424	. 2 |- ((J e. Top /\ A e. X) -> (A e. X /\ ((nei` J)` {A}) C_ ((nei` J)` {A})))
4		simpl 346	. . 3 |- ((J e. Top /\ A e. X) -> J e. Top)
5		snssi 3129	. . . . 5 |- (A e. X -> {A} C_ X)
6	5	adantl 424	. . . 4 |- ((J e. Top /\ A e. X) -> {A} C_ X)
7		snnzg 3118	. . . . 5 |- (A e. X -> {A} =/= (/))
8	7	adantl 424	. . . 4 |- ((J e. Top /\ A e. X) -> {A} =/= (/))
9		neiplim.1	. . . . 5 |- X = U.J
10	9	neifil 10302	. . . 4 |- ((J e. Top /\ {A} C_ X /\ {A} =/= (/)) -> ((nei` J)` {A}) e. Fil)
11	4, 6, 8, 10	syl111anc 1100	. . 3 |- ((J e. Top /\ A e. X) -> ((nei` J)` {A}) e. Fil)
12	9	unnei 9011	. . . . 5 |- ((J e. Top /\ {A} C_ X) -> U.((nei` J)` {A}) = X)
13	12, 5	sylan2 500	. . . 4 |- ((J e. Top /\ A e. X) -> U.((nei` J)` {A}) = X)
14	13	eqcomd 1889	. . 3 |- ((J e. Top /\ A e. X) -> X = U.((nei` J)` {A}))
15		eqid 1884	. . . 4 |- U.((nei` J)` {A}) = U.((nei` J)` {A})
16	9, 15	isfillim 10298	. . 3 |- ((J e. Top /\ ((nei` J)` {A}) e. Fil /\ X = U.((nei` J)` {A})) -> (A e. ((fLim1` J)` ((nei` J)` {A})) <-> (A e. X /\ ((nei` J)` {A}) C_ ((nei` J)` {A}))))
17	4, 11, 14, 16	syl111anc 1100	. 2 |- ((J e. Top /\ A e. X) -> (A e. ((fLim1` J)` ((nei` J)` {A})) <-> (A e. X /\ ((nei` J)` {A}) C_ ((nei` J)` {A}))))
18	3, 17	mpbird 213	1 |- ((J e. Top /\ A e. X) -> A e. ((fLim1` J)` ((nei` J)` {A})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988  Filcfil 10264  fLim1cflim1 10294

This theorem is referenced by:  flimcls 15588  cnpfillim 15589

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-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-fil 10265  df-flim1 10295

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