Theorem ufcomp 15622

Description: A space is compact iff every ultrafilter converges.

Hypothesis

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

Assertion

Ref	Expression
ufcomp	|- (J e. Top -> (J e. Comp <-> A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/))))

Distinct variable groups:   f,J   f,X

Proof of Theorem ufcomp

Step	Hyp	Ref	Expression
1		ufcomp.1	. . 3 |- X = U.J
2	1	fcluscomp 15621	. 2 |- (J e. Top -> (J e. Comp <-> A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/))))
3		simp2 877	. . . . . . . 8 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> X = U.f)
4		ufilfil 15566	. . . . . . . . . 10 |- (f e. UFil -> f e. Fil)
5		unieq 3185	. . . . . . . . . . . . 13 |- (l = f -> U.l = U.f)
6	5	eqeq2d 1895	. . . . . . . . . . . 12 |- (l = f -> (X = U.l <-> X = U.f))
7		fveq2 4681	. . . . . . . . . . . . 13 |- (l = f -> ((fClus` J)` l) = ((fClus` J)` f))
8	7	neeq1d 2028	. . . . . . . . . . . 12 |- (l = f -> (((fClus` J)` l) =/= (/) <-> ((fClus` J)` f) =/= (/)))
9	6, 8	imbi12d 688	. . . . . . . . . . 11 |- (l = f -> ((X = U.l -> ((fClus` J)` l) =/= (/)) <-> (X = U.f -> ((fClus` J)` f) =/= (/))))
10	9	rcla4v 2376	. . . . . . . . . 10 |- (f e. Fil -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> (X = U.f -> ((fClus` J)` f) =/= (/))))
11	4, 10	syl 12	. . . . . . . . 9 |- (f e. UFil -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> (X = U.f -> ((fClus` J)` f) =/= (/))))
12	11	3ad2ant3 899	. . . . . . . 8 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> (X = U.f -> ((fClus` J)` f) =/= (/))))
13	3, 12	mpid 58	. . . . . . 7 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> ((fClus` J)` f) =/= (/)))
14		eqid 1884	. . . . . . . . . 10 |- U.f = U.f
15	1, 14	uffclsflim 15616	. . . . . . . . 9 |- ((J e. Top /\ f e. UFil /\ X = U.f) -> ((fClus` J)` f) = ((fLim1` J)` f))
16	15	3com23 1074	. . . . . . . 8 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> ((fClus` J)` f) = ((fLim1` J)` f))
17	16	neeq1d 2028	. . . . . . 7 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> (((fClus` J)` f) =/= (/) <-> ((fLim1` J)` f) =/= (/)))
18	13, 17	sylibd 219	. . . . . 6 |- ((J e. Top /\ X = U.f /\ f e. UFil) -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> ((fLim1` J)` f) =/= (/)))
19	18	3exp 1066	. . . . 5 |- (J e. Top -> (X = U.f -> (f e. UFil -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> ((fLim1` J)` f) =/= (/)))))
20	19	com24 41	. . . 4 |- (J e. Top -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> (f e. UFil -> (X = U.f -> ((fLim1` J)` f) =/= (/)))))
21	20	r19.21adv 2181	. . 3 |- (J e. Top -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) -> A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/))))
22		eqid 1884	. . . . . . . . . . 11 |- U.l = U.l
23	22	filssufil 15571	. . . . . . . . . 10 |- (l e. Fil -> E.g e. UFil (U.l = U.g /\ l C_ g))
24	23	adantl 424	. . . . . . . . 9 |- ((X = U.l /\ l e. Fil) -> E.g e. UFil (U.l = U.g /\ l C_ g))
25		eqeq1 1890	. . . . . . . . . . . 12 |- (X = U.l -> (X = U.g <-> U.l = U.g))
26	25	anbi1d 679	. . . . . . . . . . 11 |- (X = U.l -> ((X = U.g /\ l C_ g) <-> (U.l = U.g /\ l C_ g)))
27	26	rexbidv 2124	. . . . . . . . . 10 |- (X = U.l -> (E.g e. UFil (X = U.g /\ l C_ g) <-> E.g e. UFil (U.l = U.g /\ l C_ g)))
28	27	adantr 425	. . . . . . . . 9 |- ((X = U.l /\ l e. Fil) -> (E.g e. UFil (X = U.g /\ l C_ g) <-> E.g e. UFil (U.l = U.g /\ l C_ g)))
29	24, 28	mpbird 213	. . . . . . . 8 |- ((X = U.l /\ l e. Fil) -> E.g e. UFil (X = U.g /\ l C_ g))
30	29	3adant1 894	. . . . . . 7 |- ((J e. Top /\ X = U.l /\ l e. Fil) -> E.g e. UFil (X = U.g /\ l C_ g))
31		simpl1 879	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> J e. Top)
32		simpr1 882	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> g e. UFil)
33		simpr2 883	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> X = U.g)
34		eqid 1884	. . . . . . . . . . . . . 14 |- U.g = U.g
35	1, 34	uffclsflim 15616	. . . . . . . . . . . . 13 |- ((J e. Top /\ g e. UFil /\ X = U.g) -> ((fClus` J)` g) = ((fLim1` J)` g))
36	31, 32, 33, 35	syl111anc 1100	. . . . . . . . . . . 12 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> ((fClus` J)` g) = ((fLim1` J)` g))
37		simpl3 881	. . . . . . . . . . . . . 14 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> l e. Fil)
38		ufilfil 15566	. . . . . . . . . . . . . . . 16 |- (g e. UFil -> g e. Fil)
39	38	3ad2ant1 897	. . . . . . . . . . . . . . 15 |- ((g e. UFil /\ X = U.g /\ l C_ g) -> g e. Fil)
40	39	adantl 424	. . . . . . . . . . . . . 14 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> g e. Fil)
41	31, 37, 40	3jca 1050	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> (J e. Top /\ l e. Fil /\ g e. Fil))
42		simpl2 880	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> X = U.l)
43		simpr3 884	. . . . . . . . . . . . 13 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> l C_ g)
44	1, 22, 34	fclusss 15611	. . . . . . . . . . . . 13 |- ((((J e. Top /\ l e. Fil /\ g e. Fil) /\ X = U.l /\ X = U.g) /\ l C_ g) -> ((fClus` J)` g) C_ ((fClus` J)` l))
45	41, 42, 33, 43, 44	syl31anc 1103	. . . . . . . . . . . 12 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> ((fClus` J)` g) C_ ((fClus` J)` l))
46	36, 45	eqsstr3d 2652	. . . . . . . . . . 11 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> ((fLim1` J)` g) C_ ((fClus` J)` l))
47		unieq 3185	. . . . . . . . . . . . . . . . 17 |- (f = g -> U.f = U.g)
48	47	eqeq2d 1895	. . . . . . . . . . . . . . . 16 |- (f = g -> (X = U.f <-> X = U.g))
49		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (f = g -> ((fLim1` J)` f) = ((fLim1` J)` g))
50	49	neeq1d 2028	. . . . . . . . . . . . . . . 16 |- (f = g -> (((fLim1` J)` f) =/= (/) <-> ((fLim1` J)` g) =/= (/)))
51	48, 50	imbi12d 688	. . . . . . . . . . . . . . 15 |- (f = g -> ((X = U.f -> ((fLim1` J)` f) =/= (/)) <-> (X = U.g -> ((fLim1` J)` g) =/= (/))))
52	51	rcla4v 2376	. . . . . . . . . . . . . 14 |- (g e. UFil -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> (X = U.g -> ((fLim1` J)` g) =/= (/))))
53	52	3ad2ant1 897	. . . . . . . . . . . . 13 |- ((g e. UFil /\ X = U.g /\ l C_ g) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> (X = U.g -> ((fLim1` J)` g) =/= (/))))
54	53	adantl 424	. . . . . . . . . . . 12 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> (X = U.g -> ((fLim1` J)` g) =/= (/))))
55	33, 54	mpid 58	. . . . . . . . . . 11 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fLim1` J)` g) =/= (/)))
56		ssn0 2905	. . . . . . . . . . . 12 |- ((((fLim1` J)` g) C_ ((fClus` J)` l) /\ ((fLim1` J)` g) =/= (/)) -> ((fClus` J)` l) =/= (/))
57	56	ex 402	. . . . . . . . . . 11 |- (((fLim1` J)` g) C_ ((fClus` J)` l) -> (((fLim1` J)` g) =/= (/) -> ((fClus` J)` l) =/= (/)))
58	46, 55, 57	sylsyld 32	. . . . . . . . . 10 |- (((J e. Top /\ X = U.l /\ l e. Fil) /\ (g e. UFil /\ X = U.g /\ l C_ g)) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/)))
59	58	3exp2 1086	. . . . . . . . 9 |- ((J e. Top /\ X = U.l /\ l e. Fil) -> (g e. UFil -> (X = U.g -> (l C_ g -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/))))))
60	59	imp4a 391	. . . . . . . 8 |- ((J e. Top /\ X = U.l /\ l e. Fil) -> (g e. UFil -> ((X = U.g /\ l C_ g) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/)))))
61	60	r19.23adv 2215	. . . . . . 7 |- ((J e. Top /\ X = U.l /\ l e. Fil) -> (E.g e. UFil (X = U.g /\ l C_ g) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/))))
62	30, 61	mpd 29	. . . . . 6 |- ((J e. Top /\ X = U.l /\ l e. Fil) -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/)))
63	62	3exp 1066	. . . . 5 |- (J e. Top -> (X = U.l -> (l e. Fil -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> ((fClus` J)` l) =/= (/)))))
64	63	com24 41	. . . 4 |- (J e. Top -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> (l e. Fil -> (X = U.l -> ((fClus` J)` l) =/= (/)))))
65	64	r19.21adv 2181	. . 3 |- (J e. Top -> (A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/)) -> A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/))))
66	21, 65	impbid 574	. 2 |- (J e. Top -> (A.l e. Fil (X = U.l -> ((fClus` J)` l) =/= (/)) <-> A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/))))
67	2, 66	bitrd 587	1 |- (J e. Top -> (J e. Comp <-> A.f e. UFil (X = U.f -> ((fLim1` J)` f) =/= (/))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  U.cuni 3177  ` cfv 3998  Topctop 8857  Filcfil 10264  fLim1cflim1 10294  Compccomp 10328  UFilcufil 15562  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  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-reu 2111  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-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-nei 8989  df-fi 10211  df-fbas 10259  df-fg 10260  df-fil 10265  df-flim1 10295  df-comp 10329  df-ufil 15563  df-fclus 15584

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