Theorem flimopn 10321

Description: The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.)

Hypotheses

Ref	Expression
flimopn.1	|- X = U.J
flimopn.2	|- Y = U.F

Assertion

Ref	Expression
flimopn	|- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A e. ((fLim1` J)` F) <-> A.o e. J (A e. o -> o e. F)))

Distinct variable groups:   A,o   o,F   o,J   o,X   o,Y

Proof of Theorem flimopn

Step	Hyp	Ref	Expression
1		flimopn.1	. . . . . . . 8 |- X = U.J
2		flimopn.2	. . . . . . . 8 |- Y = U.F
3	1, 2	flimelbas 10300	. . . . . . 7 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. ((fLim1` J)` F)) -> A e. X)
4	3	ex 402	. . . . . 6 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fLim1` J)` F) -> A e. X))
5	4	pm4.71rd 701	. . . . 5 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fLim1` J)` F) <-> (A e. X /\ A e. ((fLim1` J)` F))))
6	1, 2	isfillim 10298	. . . . 5 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fLim1` J)` F) <-> (A e. X /\ ((nei` J)` {A}) C_ F)))
7	5, 6	bitr3d 589	. . . 4 |- ((J e. Top /\ F e. Fil /\ X = Y) -> ((A e. X /\ A e. ((fLim1` J)` F)) <-> (A e. X /\ ((nei` J)` {A}) C_ F)))
8		pm5.32 706	. . . 4 |- ((A e. X -> (A e. ((fLim1` J)` F) <-> ((nei` J)` {A}) C_ F)) <-> ((A e. X /\ A e. ((fLim1` J)` F)) <-> (A e. X /\ ((nei` J)` {A}) C_ F)))
9	7, 8	sylibr 217	. . 3 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. X -> (A e. ((fLim1` J)` F) <-> ((nei` J)` {A}) C_ F)))
10	9	imp 377	. 2 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A e. ((fLim1` J)` F) <-> ((nei` J)` {A}) C_ F))
11		opnneip 9009	. . . . . . . . 9 |- ((J e. Top /\ o e. J /\ A e. o) -> o e. ((nei` J)` {A}))
12	11	3expib 1070	. . . . . . . 8 |- (J e. Top -> ((o e. J /\ A e. o) -> o e. ((nei` J)` {A})))
13	12	adantr 425	. . . . . . 7 |- ((J e. Top /\ A e. X) -> ((o e. J /\ A e. o) -> o e. ((nei` J)` {A})))
14		ssel 2615	. . . . . . 7 |- (((nei` J)` {A}) C_ F -> (o e. ((nei` J)` {A}) -> o e. F))
15	13, 14	syl9 71	. . . . . 6 |- ((J e. Top /\ A e. X) -> (((nei` J)` {A}) C_ F -> ((o e. J /\ A e. o) -> o e. F)))
16	15	exp4a 409	. . . . 5 |- ((J e. Top /\ A e. X) -> (((nei` J)` {A}) C_ F -> (o e. J -> (A e. o -> o e. F))))
17	16	3ad2antl1 1038	. . . 4 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (((nei` J)` {A}) C_ F -> (o e. J -> (A e. o -> o e. F))))
18	17	r19.21adv 2181	. . 3 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (((nei` J)` {A}) C_ F -> A.o e. J (A e. o -> o e. F)))
19	1	isnei 8994	. . . . . . . . 9 |- ((J e. Top /\ {A} C_ X) -> (n e. ((nei` J)` {A}) <-> (n C_ X /\ E.s e. J ({A} C_ s /\ s C_ n))))
20		snssi 3129	. . . . . . . . 9 |- (A e. X -> {A} C_ X)
21	19, 20	sylan2 500	. . . . . . . 8 |- ((J e. Top /\ A e. X) -> (n e. ((nei` J)` {A}) <-> (n C_ X /\ E.s e. J ({A} C_ s /\ s C_ n))))
22	21	3ad2antl1 1038	. . . . . . 7 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (n e. ((nei` J)` {A}) <-> (n C_ X /\ E.s e. J ({A} C_ s /\ s C_ n))))
23	22	adantr 425	. . . . . 6 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> (n e. ((nei` J)` {A}) <-> (n C_ X /\ E.s e. J ({A} C_ s /\ s C_ n))))
24		eleq2 1958	. . . . . . . . . . . . . . . 16 |- (o = s -> (A e. o <-> A e. s))
25		eleq1 1957	. . . . . . . . . . . . . . . 16 |- (o = s -> (o e. F <-> s e. F))
26	24, 25	imbi12d 688	. . . . . . . . . . . . . . 15 |- (o = s -> ((A e. o -> o e. F) <-> (A e. s -> s e. F)))
27	26	rcla4v 2376	. . . . . . . . . . . . . 14 |- (s e. J -> (A.o e. J (A e. o -> o e. F) -> (A e. s -> s e. F)))
28	27	adantl 424	. . . . . . . . . . . . 13 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (A.o e. J (A e. o -> o e. F) -> (A e. s -> s e. F)))
29		snssg 3124	. . . . . . . . . . . . . . . . 17 |- (A e. X -> (A e. s <-> {A} C_ s))
30	29	anbi1d 679	. . . . . . . . . . . . . . . 16 |- (A e. X -> ((A e. s /\ s C_ n) <-> ({A} C_ s /\ s C_ n)))
31	30	ad2antlr 441	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> ((A e. s /\ s C_ n) <-> ({A} C_ s /\ s C_ n)))
32		pm2.27 76	. . . . . . . . . . . . . . . . . . 19 |- (A e. s -> ((A e. s -> s e. F) -> s e. F))
33		sseq2 2639	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (X = Y -> (n C_ X <-> n C_ Y))
34	33	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((F e. Fil /\ X = Y) -> (n C_ X <-> n C_ Y))
35	2	fillsb 10270	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (F e. Fil -> ((s e. F /\ n C_ Y /\ s C_ n) -> n e. F))
36	35	3expd 1085	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (F e. Fil -> (s e. F -> (n C_ Y -> (s C_ n -> n e. F))))
37	36	com23 36	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (F e. Fil -> (n C_ Y -> (s e. F -> (s C_ n -> n e. F))))
38	37	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((F e. Fil /\ X = Y) -> (n C_ Y -> (s e. F -> (s C_ n -> n e. F))))
39	34, 38	sylbid 220	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((F e. Fil /\ X = Y) -> (n C_ X -> (s e. F -> (s C_ n -> n e. F))))
40	39	com24 41	. . . . . . . . . . . . . . . . . . . . . 22 |- ((F e. Fil /\ X = Y) -> (s C_ n -> (s e. F -> (n C_ X -> n e. F))))
41	40	3adant1 894	. . . . . . . . . . . . . . . . . . . . 21 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (s C_ n -> (s e. F -> (n C_ X -> n e. F))))
42	41	ad2antrr 440	. . . . . . . . . . . . . . . . . . . 20 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (s C_ n -> (s e. F -> (n C_ X -> n e. F))))
43	42	com13 37	. . . . . . . . . . . . . . . . . . 19 |- (s e. F -> (s C_ n -> ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (n C_ X -> n e. F))))
44	32, 43	syl6 25	. . . . . . . . . . . . . . . . . 18 |- (A e. s -> ((A e. s -> s e. F) -> (s C_ n -> ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (n C_ X -> n e. F)))))
45	44	com24 41	. . . . . . . . . . . . . . . . 17 |- (A e. s -> ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (s C_ n -> ((A e. s -> s e. F) -> (n C_ X -> n e. F)))))
46	45	com12 14	. . . . . . . . . . . . . . . 16 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (A e. s -> (s C_ n -> ((A e. s -> s e. F) -> (n C_ X -> n e. F)))))
47	46	imp3a 388	. . . . . . . . . . . . . . 15 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> ((A e. s /\ s C_ n) -> ((A e. s -> s e. F) -> (n C_ X -> n e. F))))
48	31, 47	sylbird 222	. . . . . . . . . . . . . 14 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (({A} C_ s /\ s C_ n) -> ((A e. s -> s e. F) -> (n C_ X -> n e. F))))
49	48	com23 36	. . . . . . . . . . . . 13 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> ((A e. s -> s e. F) -> (({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F))))
50	28, 49	syld 30	. . . . . . . . . . . 12 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. J) -> (A.o e. J (A e. o -> o e. F) -> (({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F))))
51	50	ex 402	. . . . . . . . . . 11 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (s e. J -> (A.o e. J (A e. o -> o e. F) -> (({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F)))))
52	51	com23 36	. . . . . . . . . 10 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> o e. F) -> (s e. J -> (({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F)))))
53	52	imp 377	. . . . . . . . 9 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> (s e. J -> (({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F))))
54	53	r19.23adv 2215	. . . . . . . 8 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> (E.s e. J ({A} C_ s /\ s C_ n) -> (n C_ X -> n e. F)))
55	54	com23 36	. . . . . . 7 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> (n C_ X -> (E.s e. J ({A} C_ s /\ s C_ n) -> n e. F)))
56	55	imp3a 388	. . . . . 6 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> ((n C_ X /\ E.s e. J ({A} C_ s /\ s C_ n)) -> n e. F))
57	23, 56	sylbid 220	. . . . 5 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> (n e. ((nei` J)` {A}) -> n e. F))
58	57	ssrdv 2622	. . . 4 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ A.o e. J (A e. o -> o e. F)) -> ((nei` J)` {A}) C_ F)
59	58	ex 402	. . 3 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> o e. F) -> ((nei` J)` {A}) C_ F))
60	18, 59	impbid 574	. 2 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (((nei` J)` {A}) C_ F <-> A.o e. J (A e. o -> o e. F)))
61	10, 60	bitrd 587	1 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A e. ((fLim1` J)` F) <-> A.o e. J (A e. o -> o e. F)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  {csn 3044  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988  Filcfil 10264  fLim1cflim1 10294

This theorem is referenced by:  fbaslim 10322  isflimf 10323  limfilcf 15587  flimfcls 15613  flimfnfcls 15615

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-nei 8989  df-fil 10265  df-flim1 10295

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