Theorem isfclus 15606

Description: A cluster point of a filter.

Hypotheses

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

Assertion

Ref	Expression
isfclus	|- ((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) =/= (/)))))

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

Proof of Theorem isfclus

Step	Hyp	Ref	Expression
1		isfclus.1	. . . 4 |- X = U.J
2		isfclus.2	. . . 4 |- Y = U.F
3	1, 2	filclus 15605	. . 3 |- ((J e. Top /\ F e. Fil /\ X = Y) -> ((fClus` J)` F) = |^|_s e. F ((cls` J)` s))
4	3	eleq2d 1964	. 2 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. ((fClus` J)` F) <-> A e. |^|_s e. F ((cls` J)` s)))
5		eleq1 1957	. . . . . . . . . 10 |- (X = Y -> (X e. F <-> Y e. F))
6	2	filusb 10267	. . . . . . . . . 10 |- (F e. Fil -> Y e. F)
7	5, 6	syl5bir 227	. . . . . . . . 9 |- (X = Y -> (F e. Fil -> X e. F))
8	7	impcom 378	. . . . . . . 8 |- ((F e. Fil /\ X = Y) -> X e. F)
9	8	3adant1 894	. . . . . . 7 |- ((J e. Top /\ F e. Fil /\ X = Y) -> X e. F)
10	1	clstop 8976	. . . . . . . . 9 |- (J e. Top -> ((cls` J)` X) = X)
11	10	3ad2ant1 897	. . . . . . . 8 |- ((J e. Top /\ F e. Fil /\ X = Y) -> ((cls` J)` X) = X)
12		eqimss 2665	. . . . . . . 8 |- (((cls` J)` X) = X -> ((cls` J)` X) C_ X)
13	11, 12	syl 12	. . . . . . 7 |- ((J e. Top /\ F e. Fil /\ X = Y) -> ((cls` J)` X) C_ X)
14		fveq2 4681	. . . . . . . . 9 |- (s = X -> ((cls` J)` s) = ((cls` J)` X))
15	14	sseq1d 2644	. . . . . . . 8 |- (s = X -> (((cls` J)` s) C_ X <-> ((cls` J)` X) C_ X))
16	15	rcla4ev 2381	. . . . . . 7 |- ((X e. F /\ ((cls` J)` X) C_ X) -> E.s e. F ((cls` J)` s) C_ X)
17	9, 13, 16	syl11anc 524	. . . . . 6 |- ((J e. Top /\ F e. Fil /\ X = Y) -> E.s e. F ((cls` J)` s) C_ X)
18		iinss 3304	. . . . . 6 |- (E.s e. F ((cls` J)` s) C_ X -> |^|_s e. F ((cls` J)` s) C_ X)
19	17, 18	syl 12	. . . . 5 |- ((J e. Top /\ F e. Fil /\ X = Y) -> |^|_s e. F ((cls` J)` s) C_ X)
20	19	sseld 2619	. . . 4 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. |^|_s e. F ((cls` J)` s) -> A e. X))
21	20	imdistani 491	. . 3 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. |^|_s e. F ((cls` J)` s)) -> ((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X))
22		simpl 346	. . . 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) =/= (/)))) -> (J e. Top /\ F e. Fil /\ X = Y))
23		simprl 450	. . . 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 e. X)
24	22, 23	jca 310	. . 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) =/= (/)))) -> ((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X))
25		eliin 3260	. . . . 5 |- (A e. X -> (A e. |^|_s e. F ((cls` J)` s) <-> A.s e. F A e. ((cls` J)` s)))
26	25	adantl 424	. . . 4 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A e. |^|_s e. F ((cls` J)` s) <-> A.s e. F A e. ((cls` J)` s)))
27		simpll1 915	. . . . . . 7 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. F) -> J e. Top)
28		sseq2 2639	. . . . . . . . . . 11 |- (X = Y -> (s C_ X <-> s C_ Y))
29	28	biimpar 461	. . . . . . . . . 10 |- ((X = Y /\ s C_ Y) -> s C_ X)
30		elssuni 3206	. . . . . . . . . . 11 |- (s e. F -> s C_ U.F)
31	30, 2	syl6ssr 2664	. . . . . . . . . 10 |- (s e. F -> s C_ Y)
32	29, 31	sylan2 500	. . . . . . . . 9 |- ((X = Y /\ s e. F) -> s C_ X)
33	32	3ad2antl3 1040	. . . . . . . 8 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ s e. F) -> s C_ X)
34	33	adantlr 429	. . . . . . 7 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. F) -> s C_ X)
35		simplr 449	. . . . . . 7 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. F) -> A e. X)
36	1	elcls 8980	. . . . . . 7 |- ((J e. Top /\ s C_ X /\ A e. X) -> (A e. ((cls` J)` s) <-> A.o e. J (A e. o -> (o i^i s) =/= (/))))
37	27, 34, 35, 36	syl111anc 1100	. . . . . 6 |- ((((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) /\ s e. F) -> (A e. ((cls` J)` s) <-> A.o e. J (A e. o -> (o i^i s) =/= (/))))
38	37	ralbidva 2119	. . . . 5 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.s e. F A e. ((cls` J)` s) <-> A.s e. F A.o e. J (A e. o -> (o i^i s) =/= (/))))
39		ralcom 2242	. . . . . 6 |- (A.s e. F A.o e. J (A e. o -> (o i^i s) =/= (/)) <-> A.o e. J A.s e. F (A e. o -> (o i^i s) =/= (/)))
40		r19.21v 2178	. . . . . . 7 |- (A.s e. F (A e. o -> (o i^i s) =/= (/)) <-> (A e. o -> A.s e. F (o i^i s) =/= (/)))
41	40	ralbii 2127	. . . . . 6 |- (A.o e. J A.s e. F (A e. o -> (o i^i s) =/= (/)) <-> A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))
42	39, 41	bitri 190	. . . . 5 |- (A.s e. F A.o e. J (A e. o -> (o i^i s) =/= (/)) <-> A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))
43	38, 42	syl6bb 595	. . . 4 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A.s e. F A e. ((cls` J)` s) <-> A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/))))
44		ibar 705	. . . . 5 |- (A e. X -> (A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)) <-> (A e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))))
45	44	adantl 424	. . . 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 e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))))
46	26, 43, 45	3bitrd 603	. . 3 |- (((J e. Top /\ F e. Fil /\ X = Y) /\ A e. X) -> (A e. |^|_s e. F ((cls` J)` s) <-> (A e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))))
47	21, 24, 46	pm5.21nd 744	. 2 |- ((J e. Top /\ F e. Fil /\ X = Y) -> (A e. |^|_s e. F ((cls` J)` s) <-> (A e. X /\ A.o e. J (A e. o -> A.s e. F (o i^i s) =/= (/)))))
48	4, 47	bitrd 587	1 |- ((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) =/= (/)))))

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  |^|_ciin 3256  ` cfv 3998  Topctop 8857  clsccl 8938  Filcfil 10264  fCluscfclus 15582

This theorem is referenced by:  fclusnei 15607  fclselbas 15608  fclusbas 15610  fcluscf 15612  flimfcls 15613  flimfnfcls 15615  isfclusf 15625

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-fil 10265  df-fclus 15584

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