Theorem fclusnei 15607

Description: Cluster points in terms of neighborhoods.

Hypotheses

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

Assertion

Ref	Expression
fclusnei	|- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/))))

Distinct variable groups:   n,s,A   n,F,s   n,J,s   n,X,s   n,Y,s

Proof of Theorem fclusnei

Step	Hyp	Ref	Expression
1		fclusnei.1	. . 3 |- X = U.J
2		fclusnei.2	. . 3 |- Y = U.F
3	1, 2	isfclus 15606	. 2 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))))
4		neii2 8998	. . . . . . . . . 10 |- ((J e. Top /\ n e. ((nei` J)` {A})) -> E.p e. J ({A} C_ p /\ p C_ n))
5	4	ex 402	. . . . . . . . 9 |- (J e. Top -> (n e. ((nei` J)` {A}) -> E.p e. J ({A} C_ p /\ p C_ n)))
6	5	3ad2ant1 897	. . . . . . . 8 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (n e. ((nei` J)` {A}) -> E.p e. J ({A} C_ p /\ p C_ n)))
7	6	adantr 425	. . . . . . 7 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (n e. ((nei` J)` {A}) -> E.p e. J ({A} C_ p /\ p C_ n)))
8		snssg 3124	. . . . . . . . . . . . . 14 |- (A e. X -> (A e. p <-> {A} C_ p))
9	8	biimpar 461	. . . . . . . . . . . . 13 |- ((A e. X /\ {A} C_ p) -> A e. p)
10	9	adantrr 431	. . . . . . . . . . . 12 |- ((A e. X /\ ({A} C_ p /\ p C_ n)) -> A e. p)
11	10	ad2ant2l 444	. . . . . . . . . . 11 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) -> A e. p)
12		eleq2 1958	. . . . . . . . . . . . . 14 |- (o = p -> (A e. o <-> A e. p))
13		ineq1 2789	. . . . . . . . . . . . . . . 16 |- (o = p -> (o i^i s) = (p i^i s))
14	13	neeq1d 2028	. . . . . . . . . . . . . . 15 |- (o = p -> ((o i^i s) =/= (/) <-> (p i^i s) =/= (/)))
15	14	ralbidv 2123	. . . . . . . . . . . . . 14 |- (o = p -> (A.s e. F (o i^i s) =/= (/) <-> A.s e. F (p i^i s) =/= (/)))
16	12, 15	imbi12d 688	. . . . . . . . . . . . 13 |- (o = p -> ((A e. o -> A.s e. F (o i^i s) =/= (/)) <-> (A e. p -> A.s e. F (p i^i s) =/= (/))))
17	16	rcla4v 2376	. . . . . . . . . . . 12 |- (p e. J -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> (A e. p -> A.s e. F (p i^i s) =/= (/))))
18	17	ad2antrl 442	. . . . . . . . . . 11 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> (A e. p -> A.s e. F (p i^i s) =/= (/))))
19	11, 18	mpid 58	. . . . . . . . . 10 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.s e. F (p i^i s) =/= (/)))
20		ssrin 2817	. . . . . . . . . . . . . 14 |- (p C_ n -> (p i^i s) C_ (n i^i s))
21	20	ad2antll 443	. . . . . . . . . . . . 13 |- ((p e. J /\ ({A} C_ p /\ p C_ n)) -> (p i^i s) C_ (n i^i s))
22	21	ad2antlr 441	. . . . . . . . . . . 12 |- (((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) /\ s e. F) -> (p i^i s) C_ (n i^i s))
23		ssn0 2905	. . . . . . . . . . . . 13 |- (((p i^i s) C_ (n i^i s) /\ (p i^i s) =/= (/)) -> (n i^i s) =/= (/))
24	23	ex 402	. . . . . . . . . . . 12 |- ((p i^i s) C_ (n i^i s) -> ((p i^i s) =/= (/) -> (n i^i s) =/= (/)))
25	22, 24	syl 12	. . . . . . . . . . 11 |- (((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) /\ s e. F) -> ((p i^i s) =/= (/) -> (n i^i s) =/= (/)))
26	25	ralimdvaa 2171	. . . . . . . . . 10 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) -> (A.s e. F (p i^i s) =/= (/) -> A.s e. F (n i^i s) =/= (/)))
27	19, 26	syld 30	. . . . . . . . 9 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (p e. J /\ ({A} C_ p /\ p C_ n))) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.s e. F (n i^i s) =/= (/)))
28	27	exp32 408	. . . . . . . 8 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (p e. J -> (({A} C_ p /\ p C_ n) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.s e. F (n i^i s) =/= (/)))))
29	28	r19.23adv 2215	. . . . . . 7 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (E.p e. J ({A} C_ p /\ p C_ n) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.s e. F (n i^i s) =/= (/))))
30	7, 29	syld 30	. . . . . 6 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (n e. ((nei` J)` {A}) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.s e. F (n i^i s) =/= (/))))
31	30	com23 36	. . . . 5 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> (n e. ((nei` J)` {A}) -> A.s e. F (n i^i s) =/= (/))))
32	31	r19.21adv 2181	. . . 4 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) -> A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/)))
33		opnneip 9009	. . . . . . . . . . . 12 |- ((J e. Top /\ o e. J /\ A e. o) -> o e. ((nei` J)` {A}))
34	33	3expb 1068	. . . . . . . . . . 11 |- ((J e. Top /\ (o e. J /\ A e. o)) -> o e. ((nei` J)` {A}))
35	34	3ad2antl1 1038	. . . . . . . . . 10 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ (o e. J /\ A e. o)) -> o e. ((nei` J)` {A}))
36	35	adantlr 429	. . . . . . . . 9 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (o e. J /\ A e. o)) -> o e. ((nei` J)` {A}))
37		ineq1 2789	. . . . . . . . . . . 12 |- (n = o -> (n i^i s) = (o i^i s))
38	37	neeq1d 2028	. . . . . . . . . . 11 |- (n = o -> ((n i^i s) =/= (/) <-> (o i^i s) =/= (/)))
39	38	ralbidv 2123	. . . . . . . . . 10 |- (n = o -> (A.s e. F (n i^i s) =/= (/) <-> A.s e. F (o i^i s) =/= (/)))
40	39	rcla4v 2376	. . . . . . . . 9 |- (o e. ((nei` J)` {A}) -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> A.s e. F (o i^i s) =/= (/)))
41	36, 40	syl 12	. . . . . . . 8 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ (o e. J /\ A e. o)) -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> A.s e. F (o i^i s) =/= (/)))
42	41	exp32 408	. . . . . . 7 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (o e. J -> (A e. o -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> A.s e. F (o i^i s) =/= (/)))))
43	42	com34 40	. . . . . 6 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (o e. J -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> (A e. o -> A.s e. F (o i^i s) =/= (/)))))
44	43	com23 36	. . . . 5 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> (o e. J -> (A e. o -> A.s e. F (o i^i s) =/= (/)))))
45	44	r19.21adv 2181	. . . 4 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/) -> A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/))))
46	32, 45	impbid 574	. . 3 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) <-> A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/)))
47	46	pm5.32da 711	. 2 |- ((J e. Top /\ F e. Fil /\ X = Y) -> ((A e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/))) <-> (A e. X /\ A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/))))
48	3, 47	bitrd 587	1 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.n e. ((nei` J)` {A})A.s e. F (n i^i s) =/= (/))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988  Filcfil 10264  fCluscfclus 15582

This theorem is referenced by:  fclusneii 15609

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-sbc 2454  df-csb 2541  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-int 3215  df-iun 3257  df-iin 3258  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-cld 8939  df-ntr 8940  df-cls 8941  df-nei 8989  df-fil 10265  df-fclus 15584

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