Theorem fclusbas 15610

Description: Cluster points in terms of filter bases.

Hypotheses

Ref	Expression
fclusbas.1	|- X = U.J
fclusbas.2	|- Y = U.B
fclusbas.3	|- F = (filGen` B)

Assertion

Ref	Expression
fclusbas	|- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A e. ((fClus` J)` F) <-> A.o e. J (A e. o -> A.s e. B (o i^i s) =/= (/))))

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

Proof of Theorem fclusbas

Step	Hyp	Ref	Expression
1		simp1 876	. . . 4 |- ((J e. Top /\ B e. fBas /\ X = Y) -> J e. Top)
2		fgfil 10290	. . . . . 6 |- (B e. fBas -> (filGen` B) e. Fil)
3		fclusbas.3	. . . . . 6 |- F = (filGen` B)
4	2, 3	syl5eqel 1975	. . . . 5 |- (B e. fBas -> F e. Fil)
5	4	3ad2ant2 898	. . . 4 |- ((J e. Top /\ B e. fBas /\ X = Y) -> F e. Fil)
6		fclusbas.2	. . . . . . . . 9 |- Y = U.B
7	6	fgbas 10286	. . . . . . . 8 |- (B e. fBas -> Y = U.(filGen` B))
8	3	unieqi 3187	. . . . . . . 8 |- U.F = U.(filGen` B)
9	7, 8	syl6eqr 1946	. . . . . . 7 |- (B e. fBas -> Y = U.F)
10	9	eqeq2d 1895	. . . . . 6 |- (B e. fBas -> (X = Y <-> X = U.F))
11	10	biimpa 460	. . . . 5 |- ((B e. fBas /\ X = Y) -> X = U.F)
12	11	3adant1 894	. . . 4 |- ((J e. Top /\ B e. fBas /\ X = Y) -> X = U.F)
13		fclusbas.1	. . . . 5 |- X = U.J
14		eqid 1884	. . . . 5 |- U.F = U.F
15	13, 14	isfclus 15606	. . . 4 |- ((J e. Top /\ F e. Fil /\ X = U.F) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)))))
16	1, 5, 12, 15	syl111anc 1100	. . 3 |- ((J e. Top /\ B e. fBas /\ X = Y) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)))))
17	16	adantr 425	. 2 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A e. ((fClus` J)` F) <-> (A e. X /\ A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)))))
18		ibar 705	. . 3 |- (A e. X -> (A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)) <-> (A e. X /\ A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)))))
19	18	adantl 424	. 2 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)) <-> (A e. X /\ A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)))))
20		fbssfg 10285	. . . . . . . . . . . 12 |- (B e. fBas -> B C_ (filGen` B))
21	20	sseld 2619	. . . . . . . . . . 11 |- (B e. fBas -> (s e. B -> s e. (filGen` B)))
22	21	imp 377	. . . . . . . . . 10 |- ((B e. fBas /\ s e. B) -> s e. (filGen` B))
23	22, 3	syl6eleqr 1982	. . . . . . . . 9 |- ((B e. fBas /\ s e. B) -> s e. F)
24		ineq2 2790	. . . . . . . . . . 11 |- (t = s -> (o i^i t) = (o i^i s))
25	24	neeq1d 2028	. . . . . . . . . 10 |- (t = s -> ((o i^i t) =/= (/) <-> (o i^i s) =/= (/)))
26	25	rcla4v 2376	. . . . . . . . 9 |- (s e. F -> (A.t e. F (o i^i t) =/= (/) -> (o i^i s) =/= (/)))
27	23, 26	syl 12	. . . . . . . 8 |- ((B e. fBas /\ s e. B) -> (A.t e. F (o i^i t) =/= (/) -> (o i^i s) =/= (/)))
28	27	r19.21adva 2182	. . . . . . 7 |- (B e. fBas -> (A.t e. F (o i^i t) =/= (/) -> A.s e. B (o i^i s) =/= (/)))
29	6	elfg 10284	. . . . . . . . . . 11 |- (B e. fBas -> (t e. (filGen` B) <-> (t C_ Y /\ E.r e. B r C_ t)))
30	3	eleq2i 1961	. . . . . . . . . . 11 |- (t e. F <-> t e. (filGen` B))
31	29, 30	syl5bb 591	. . . . . . . . . 10 |- (B e. fBas -> (t e. F <-> (t C_ Y /\ E.r e. B r C_ t)))
32		ineq2 2790	. . . . . . . . . . . . . . . 16 |- (s = r -> (o i^i s) = (o i^i r))
33	32	neeq1d 2028	. . . . . . . . . . . . . . 15 |- (s = r -> ((o i^i s) =/= (/) <-> (o i^i r) =/= (/)))
34	33	rcla4v 2376	. . . . . . . . . . . . . 14 |- (r e. B -> (A.s e. B (o i^i s) =/= (/) -> (o i^i r) =/= (/)))
35		ssn0 2905	. . . . . . . . . . . . . . . 16 |- (((o i^i r) C_ (o i^i t) /\ (o i^i r) =/= (/)) -> (o i^i t) =/= (/))
36		sslin 2819	. . . . . . . . . . . . . . . 16 |- (r C_ t -> (o i^i r) C_ (o i^i t))
37	35, 36	sylan 497	. . . . . . . . . . . . . . 15 |- ((r C_ t /\ (o i^i r) =/= (/)) -> (o i^i t) =/= (/))
38	37	expcom 403	. . . . . . . . . . . . . 14 |- ((o i^i r) =/= (/) -> (r C_ t -> (o i^i t) =/= (/)))
39	34, 38	syl6 25	. . . . . . . . . . . . 13 |- (r e. B -> (A.s e. B (o i^i s) =/= (/) -> (r C_ t -> (o i^i t) =/= (/))))
40	39	com23 36	. . . . . . . . . . . 12 |- (r e. B -> (r C_ t -> (A.s e. B (o i^i s) =/= (/) -> (o i^i t) =/= (/))))
41	40	r19.23aiv 2211	. . . . . . . . . . 11 |- (E.r e. B r C_ t -> (A.s e. B (o i^i s) =/= (/) -> (o i^i t) =/= (/)))
42	41	adantl 424	. . . . . . . . . 10 |- ((t C_ Y /\ E.r e. B r C_ t) -> (A.s e. B (o i^i s) =/= (/) -> (o i^i t) =/= (/)))
43	31, 42	syl6bi 231	. . . . . . . . 9 |- (B e. fBas -> (t e. F -> (A.s e. B (o i^i s) =/= (/) -> (o i^i t) =/= (/))))
44	43	com23 36	. . . . . . . 8 |- (B e. fBas -> (A.s e. B (o i^i s) =/= (/) -> (t e. F -> (o i^i t) =/= (/))))
45	44	r19.21adv 2181	. . . . . . 7 |- (B e. fBas -> (A.s e. B (o i^i s) =/= (/) -> A.t e. F (o i^i t) =/= (/)))
46	28, 45	impbid 574	. . . . . 6 |- (B e. fBas -> (A.t e. F (o i^i t) =/= (/) <-> A.s e. B (o i^i s) =/= (/)))
47	46	3ad2ant2 898	. . . . 5 |- ((J e. Top /\ B e. fBas /\ X = Y) -> (A.t e. F (o i^i t) =/= (/) <-> A.s e. B (o i^i s) =/= (/)))
48	47	adantr 425	. . . 4 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A.t e. F (o i^i t) =/= (/) <-> A.s e. B (o i^i s) =/= (/)))
49	48	imbi2d 674	. . 3 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> ((A e. o -> A.t e. F (o i^i t) =/= (/)) <-> (A e. o -> A.s e. B (o i^i s) =/= (/))))
50	49	ralbidv 2123	. 2 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A.o e. J (A e. o -> A.t e. F (o i^i t) =/= (/)) <-> A.o e. J (A e. o -> A.s e. B (o i^i s) =/= (/))))
51	17, 19, 50	3bitr2d 605	1 |- (((J e. Top /\ B e. fBas /\ X = Y) /\ A e. X) -> (A e. ((fClus` J)` F) <-> A.o e. J (A e. o -> A.s e. B (o 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  U.cuni 3177  ` cfv 3998  Topctop 8857  fBascfbas 10257  filGencfg 10258  Filcfil 10264  fCluscfclus 15582

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-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-fbas 10259  df-fg 10260  df-fil 10265  df-fclus 15584

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