Theorem fcluscomp 15621

Description: A space is compact iff every filter clusters.

Hypothesis

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

Assertion

Ref	Expression
fcluscomp	|- (J e. Top -> (J e. Comp <-> A.f e. Fil (X = U.f -> ((fClus` J)` f) =/= (/))))

Distinct variable groups:   f,J   f,X

Proof of Theorem fcluscomp

Step	Hyp	Ref	Expression
1		filesn 10268	. . . . . . . . . . . 12 |- (f e. Fil -> -. (/) e. f)
2	1	3ad2ant2 898	. . . . . . . . . . 11 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> -. (/) e. f)
3		fcluscomp.1	. . . . . . . . . . . . 13 |- X = U.J
4	3	fcluscomplem 15620	. . . . . . . . . . . 12 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> A.x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)})E.z e. f z C_ x)
5		sseq2 2639	. . . . . . . . . . . . . . 15 |- (x = (/) -> (z C_ x <-> z C_ (/)))
6	5	rexbidv 2124	. . . . . . . . . . . . . 14 |- (x = (/) -> (E.z e. f z C_ x <-> E.z e. f z C_ (/)))
7	6	rcla4cv 2377	. . . . . . . . . . . . 13 |- (A.x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)})E.z e. f z C_ x -> ((/) e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> E.z e. f z C_ (/)))
8		ss0 2902	. . . . . . . . . . . . . . . . 17 |- (z C_ (/) -> z = (/))
9	8	adantl 424	. . . . . . . . . . . . . . . 16 |- ((z e. f /\ z C_ (/)) -> z = (/))
10		simpl 346	. . . . . . . . . . . . . . . 16 |- ((z e. f /\ z C_ (/)) -> z e. f)
11	9, 10	eqeltrrd 1972	. . . . . . . . . . . . . . 15 |- ((z e. f /\ z C_ (/)) -> (/) e. f)
12	11	r19.23aiva 2212	. . . . . . . . . . . . . 14 |- (E.z e. f z C_ (/) -> (/) e. f)
13	12	a1i 8	. . . . . . . . . . . . 13 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (E.z e. f z C_ (/) -> (/) e. f))
14	7, 13	syl9r 72	. . . . . . . . . . . 12 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (A.x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)})E.z e. f z C_ x -> ((/) e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> (/) e. f)))
15	4, 14	mpd 29	. . . . . . . . . . 11 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> ((/) e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> (/) e. f))
16	2, 15	mtod 123	. . . . . . . . . 10 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> -. (/) e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}))
17		df-nel 2020	. . . . . . . . . 10 |- ((/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)}) <-> -. (/) e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}))
18	16, 17	sylibr 217	. . . . . . . . 9 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)}))
19		eleq1 1957	. . . . . . . . . . . . . 14 |- (y = ((cls` J)` s) -> (y e. (Clsd` J) <-> ((cls` J)` s) e. (Clsd` J)))
20		simpl1 879	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) -> J e. Top)
21		elssuni 3206	. . . . . . . . . . . . . . . . . 18 |- (s e. f -> s C_ U.f)
22	21	adantl 424	. . . . . . . . . . . . . . . . 17 |- ((X = U.f /\ s e. f) -> s C_ U.f)
23		simpl 346	. . . . . . . . . . . . . . . . 17 |- ((X = U.f /\ s e. f) -> X = U.f)
24	22, 23	sseqtr4d 2654	. . . . . . . . . . . . . . . 16 |- ((X = U.f /\ s e. f) -> s C_ X)
25	24	3ad2antl3 1040	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) -> s C_ X)
26	3	clscld 8959	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ s C_ X) -> ((cls` J)` s) e. (Clsd` J))
27	20, 25, 26	syl11anc 524	. . . . . . . . . . . . . 14 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) -> ((cls` J)` s) e. (Clsd` J))
28	19, 27	syl5cbir 228	. . . . . . . . . . . . 13 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) -> (y = ((cls` J)` s) -> y e. (Clsd` J)))
29	28	r19.23adva 2216	. . . . . . . . . . . 12 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (E.s e. f y = ((cls` J)` s) -> y e. (Clsd` J)))
30	29	19.21aiv 1664	. . . . . . . . . . 11 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> A.y(E.s e. f y = ((cls` J)` s) -> y e. (Clsd` J)))
31		fvex 4689	. . . . . . . . . . . . 13 |- (Clsd` J) e. _V
32	31	elpw2 3464	. . . . . . . . . . . 12 |- ({y | E.s e. f y = ((cls` J)` s)} e. ~P(Clsd` J) <-> {y | E.s e. f y = ((cls` J)` s)} C_ (Clsd` J))
33		abss 2676	. . . . . . . . . . . 12 |- ({y | E.s e. f y = ((cls` J)` s)} C_ (Clsd` J) <-> A.y(E.s e. f y = ((cls` J)` s) -> y e. (Clsd` J)))
34	32, 33	bitr2i 191	. . . . . . . . . . 11 |- (A.y(E.s e. f y = ((cls` J)` s) -> y e. (Clsd` J)) <-> {y | E.s e. f y = ((cls` J)` s)} e. ~P(Clsd` J))
35	30, 34	sylib 215	. . . . . . . . . 10 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> {y | E.s e. f y = ((cls` J)` s)} e. ~P(Clsd` J))
36		fveq2 4681	. . . . . . . . . . . . 13 |- (c = {y | E.s e. f y = ((cls` J)` s)} -> ( fi ` c) = ( fi ` {y | E.s e. f y = ((cls` J)` s)}))
37		neleq2 2102	. . . . . . . . . . . . 13 |- (( fi ` c) = ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> ((/) e/ ( fi ` c) <-> (/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)})))
38	36, 37	syl 12	. . . . . . . . . . . 12 |- (c = {y | E.s e. f y = ((cls` J)` s)} -> ((/) e/ ( fi ` c) <-> (/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)})))
39		inteq 3217	. . . . . . . . . . . . 13 |- (c = {y | E.s e. f y = ((cls` J)` s)} -> |^|c = |^|{y | E.s e. f y = ((cls` J)` s)})
40	39	neeq1d 2028	. . . . . . . . . . . 12 |- (c = {y | E.s e. f y = ((cls` J)` s)} -> (|^|c =/= (/) <-> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/)))
41	38, 40	imbi12d 688	. . . . . . . . . . 11 |- (c = {y | E.s e. f y = ((cls` J)` s)} -> (((/) e/ ( fi ` c) -> |^|c =/= (/)) <-> ((/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/))))
42	41	rcla4v 2376	. . . . . . . . . 10 |- ({y | E.s e. f y = ((cls` J)` s)} e. ~P(Clsd` J) -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> ((/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/))))
43	35, 42	syl 12	. . . . . . . . 9 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> ((/) e/ ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/))))
44	18, 43	mpid 58	. . . . . . . 8 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/)))
45		fvex 4689	. . . . . . . . . 10 |- ((cls` J)` s) e. _V
46	45	dfiin2 3287	. . . . . . . . 9 |- |^|_s e. f ((cls` J)` s) = |^|{y | E.s e. f y = ((cls` J)` s)}
47	46	neeq1i 2026	. . . . . . . 8 |- (|^|_s e. f ((cls` J)` s) =/= (/) <-> |^|{y | E.s e. f y = ((cls` J)` s)} =/= (/))
48	44, 47	syl6ibr 230	. . . . . . 7 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> |^|_s e. f ((cls` J)` s) =/= (/)))
49	48	3exp 1066	. . . . . 6 |- (J e. Top -> (f e. Fil -> (X = U.f -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> |^|_s e. f ((cls` J)` s) =/= (/)))))
50	49	com34 40	. . . . 5 |- (J e. Top -> (f e. Fil -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)))))
51	50	com23 36	. . . 4 |- (J e. Top -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> (f e. Fil -> (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)))))
52	51	r19.21adv 2181	. . 3 |- (J e. Top -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) -> A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/))))
53		vn0 2882	. . . . . . . . . . . 12 |- _V =/= (/)
54		int0 3230	. . . . . . . . . . . . 13 |- |^|(/) = _V
55	54	neeq1i 2026	. . . . . . . . . . . 12 |- (|^|(/) =/= (/) <-> _V =/= (/))
56	53, 55	mpbir 207	. . . . . . . . . . 11 |- |^|(/) =/= (/)
57		inteq 3217	. . . . . . . . . . . 12 |- (c = (/) -> |^|c = |^|(/))
58	57	neeq1d 2028	. . . . . . . . . . 11 |- (c = (/) -> (|^|c =/= (/) <-> |^|(/) =/= (/)))
59	56, 58	mpbiri 211	. . . . . . . . . 10 |- (c = (/) -> |^|c =/= (/))
60	59	a1i13 15326	. . . . . . . . 9 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> (c = (/) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/))))
61		ssel 2615	. . . . . . . . . . . . . . . . . . . 20 |- (c C_ (Clsd` J) -> (s e. c -> s e. (Clsd` J)))
62	61	3ad2ant2 898	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> (s e. c -> s e. (Clsd` J)))
63	62	adantr 425	. . . . . . . . . . . . . . . . . 18 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (s e. c -> s e. (Clsd` J)))
64	3	cldss 8947	. . . . . . . . . . . . . . . . . . . . 21 |- ((J e. Top /\ s e. (Clsd` J)) -> s C_ X)
65	64	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- (J e. Top -> (s e. (Clsd` J) -> s C_ X))
66	65	3ad2ant1 897	. . . . . . . . . . . . . . . . . . 19 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> (s e. (Clsd` J) -> s C_ X))
67	66	adantr 425	. . . . . . . . . . . . . . . . . 18 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (s e. (Clsd` J) -> s C_ X))
68	63, 67	syld 30	. . . . . . . . . . . . . . . . 17 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (s e. c -> s C_ X))
69	68	r19.21aiv 2175	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> A.s e. c s C_ X)
70		unissb 3208	. . . . . . . . . . . . . . . 16 |- (U.c C_ X <-> A.s e. c s C_ X)
71	69, 70	sylibr 217	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> U.c C_ X)
72		visset 2295	. . . . . . . . . . . . . . . 16 |- c e. _V
73		fiuni 10219	. . . . . . . . . . . . . . . 16 |- (c e. _V -> U.c = U.( fi ` c))
74	72, 73	ax-mp 7	. . . . . . . . . . . . . . 15 |- U.c = U.( fi ` c)
75	71, 74	syl5ssr 2662	. . . . . . . . . . . . . 14 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> U.( fi ` c) C_ X)
76		uniexg 3795	. . . . . . . . . . . . . . . . 17 |- (J e. Top -> U.J e. _V)
77	76, 3	syl5eqel 1975	. . . . . . . . . . . . . . . 16 |- (J e. Top -> X e. _V)
78	77	3ad2ant1 897	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> X e. _V)
79	78	adantr 425	. . . . . . . . . . . . . 14 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> X e. _V)
80		eqid 1884	. . . . . . . . . . . . . . 15 |- U.( fi ` c) = U.( fi ` c)
81	80	extbas2 10292	. . . . . . . . . . . . . 14 |- ((U.( fi ` c) C_ X /\ X e. _V) -> U.(( fi ` c) u. {X}) = X)
82	75, 79, 81	syl11anc 524	. . . . . . . . . . . . 13 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> U.(( fi ` c) u. {X}) = X)
83		pm3.22 486	. . . . . . . . . . . . . . . . 17 |- (((/) e/ ( fi ` c) /\ c =/= (/)) -> (c =/= (/) /\ (/) e/ ( fi ` c)))
84	83	3ad2antl3 1040	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (c =/= (/) /\ (/) e/ ( fi ` c)))
85		fsubbas 10281	. . . . . . . . . . . . . . . . 17 |- (c e. _V -> (( fi ` c) e. fBas <-> (c =/= (/) /\ (/) e/ ( fi ` c))))
86	72, 85	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (( fi ` c) e. fBas <-> (c =/= (/) /\ (/) e/ ( fi ` c)))
87	84, 86	sylibr 217	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> ( fi ` c) e. fBas)
88	80	extbas1 10291	. . . . . . . . . . . . . . 15 |- ((( fi ` c) e. fBas /\ U.( fi ` c) C_ X) -> (( fi ` c) u. {X}) e. fBas)
89	87, 75, 88	syl11anc 524	. . . . . . . . . . . . . 14 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (( fi ` c) u. {X}) e. fBas)
90		eqid 1884	. . . . . . . . . . . . . . 15 |- U.(( fi ` c) u. {X}) = U.(( fi ` c) u. {X})
91	90	fgbas 10286	. . . . . . . . . . . . . 14 |- ((( fi ` c) u. {X}) e. fBas -> U.(( fi ` c) u. {X}) = U.(filGen` (( fi ` c) u. {X})))
92	89, 91	syl 12	. . . . . . . . . . . . 13 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> U.(( fi ` c) u. {X}) = U.(filGen` (( fi ` c) u. {X})))
93	82, 92	eqtr3d 1927	. . . . . . . . . . . 12 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> X = U.(filGen` (( fi ` c) u. {X})))
94		fgfil 10290	. . . . . . . . . . . . 13 |- ((( fi ` c) u. {X}) e. fBas -> (filGen` (( fi ` c) u. {X})) e. Fil)
95		unieq 3185	. . . . . . . . . . . . . . . 16 |- (f = (filGen` (( fi ` c) u. {X})) -> U.f = U.(filGen` (( fi ` c) u. {X})))
96	95	eqeq2d 1895	. . . . . . . . . . . . . . 15 |- (f = (filGen` (( fi ` c) u. {X})) -> (X = U.f <-> X = U.(filGen` (( fi ` c) u. {X}))))
97		iineq1 3270	. . . . . . . . . . . . . . . 16 |- (f = (filGen` (( fi ` c) u. {X})) -> |^|_s e. f ((cls` J)` s) = |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s))
98	97	neeq1d 2028	. . . . . . . . . . . . . . 15 |- (f = (filGen` (( fi ` c) u. {X})) -> (|^|_s e. f ((cls` J)` s) =/= (/) <-> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/)))
99	96, 98	imbi12d 688	. . . . . . . . . . . . . 14 |- (f = (filGen` (( fi ` c) u. {X})) -> ((X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) <-> (X = U.(filGen` (( fi ` c) u. {X})) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/))))
100	99	rcla4v 2376	. . . . . . . . . . . . 13 |- ((filGen` (( fi ` c) u. {X})) e. Fil -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> (X = U.(filGen` (( fi ` c) u. {X})) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/))))
101	89, 94, 100	3syl 24	. . . . . . . . . . . 12 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> (X = U.(filGen` (( fi ` c) u. {X})) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/))))
102	93, 101	mpid 58	. . . . . . . . . . 11 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/)))
103		sseq2 2639	. . . . . . . . . . . . . 14 |- (|^|c = (/) -> (|^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) C_ |^|c <-> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) C_ (/)))
104		abfi2 10216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (c e. _V -> c C_ ( fi ` c))
105	72, 104	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- c C_ ( fi ` c)
106	105	a1i 8	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> c C_ ( fi ` c))
107		ssun1 2767	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( fi ` c) C_ (( fi ` c) u. {X})
108	107	a1i 8	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> ( fi ` c) C_ (( fi ` c) u. {X}))
109	106, 108	sstrd 2627	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> c C_ (( fi ` c) u. {X}))
110		fbssfg 10285	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((( fi ` c) u. {X}) e. fBas -> (( fi ` c) u. {X}) C_ (filGen` (( fi ` c) u. {X})))
111	89, 110	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (( fi ` c) u. {X}) C_ (filGen` (( fi ` c) u. {X})))
112	109, 111	sstrd 2627	. . . . . . . . . . . . . . . . . . . . . 22 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> c C_ (filGen` (( fi ` c) u. {X})))
113	112	sseld 2619	. . . . . . . . . . . . . . . . . . . . 21 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (x e. c -> x e. (filGen` (( fi ` c) u. {X}))))
114	113	impr 422	. . . . . . . . . . . . . . . . . . . 20 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ (c =/= (/) /\ x e. c)) -> x e. (filGen` (( fi ` c) u. {X})))
115		cldcls 8958	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((J e. Top /\ x e. (Clsd` J)) -> ((cls` J)` x) = x)
116		ssel2 2616	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((c C_ (Clsd` J) /\ x e. c) -> x e. (Clsd` J))
117	115, 116	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((J e. Top /\ (c C_ (Clsd` J) /\ x e. c)) -> ((cls` J)` x) = x)
118	117	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((J e. Top /\ c C_ (Clsd` J)) /\ x e. c) -> ((cls` J)` x) = x)
119	118	3adantl3 1034	. . . . . . . . . . . . . . . . . . . . . 22 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ x e. c) -> ((cls` J)` x) = x)
120	119	adantrl 430	. . . . . . . . . . . . . . . . . . . . 21 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ (c =/= (/) /\ x e. c)) -> ((cls` J)` x) = x)
121	120	eqcomd 1889	. . . . . . . . . . . . . . . . . . . 20 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ (c =/= (/) /\ x e. c)) -> x = ((cls` J)` x))
122		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . 22 |- (s = x -> ((cls` J)` s) = ((cls` J)` x))
123	122	eqeq2d 1895	. . . . . . . . . . . . . . . . . . . . 21 |- (s = x -> (x = ((cls` J)` s) <-> x = ((cls` J)` x)))
124	123	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. (filGen` (( fi ` c) u. {X})) /\ x = ((cls` J)` x)) -> E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s))
125	114, 121, 124	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ (c =/= (/) /\ x e. c)) -> E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s))
126	125	expr 418	. . . . . . . . . . . . . . . . . 18 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (x e. c -> E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)))
127	126	19.21aiv 1664	. . . . . . . . . . . . . . . . 17 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> A.x(x e. c -> E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)))
128		ssab 2677	. . . . . . . . . . . . . . . . 17 |- (c C_ {x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)} <-> A.x(x e. c -> E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)))
129	127, 128	sylibr 217	. . . . . . . . . . . . . . . 16 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> c C_ {x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)})
130		intss 3239	. . . . . . . . . . . . . . . 16 |- (c C_ {x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)} -> |^|{x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)} C_ |^|c)
131	129, 130	syl 12	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> |^|{x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)} C_ |^|c)
132	45	dfiin2 3287	. . . . . . . . . . . . . . 15 |- |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) = |^|{x | E.s e. (filGen` (( fi ` c) u. {X}))x = ((cls` J)` s)}
133	131, 132	syl5ss 2661	. . . . . . . . . . . . . 14 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) C_ |^|c)
134	103, 133	syl5cbi 226	. . . . . . . . . . . . 13 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (|^|c = (/) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) C_ (/)))
135		ss0 2902	. . . . . . . . . . . . 13 |- (|^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) C_ (/) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) = (/))
136	134, 135	syl6 25	. . . . . . . . . . . 12 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (|^|c = (/) -> |^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) = (/)))
137	136	necon3d 2041	. . . . . . . . . . 11 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (|^|_s e. (filGen` (( fi ` c) u. {X}))((cls` J)` s) =/= (/) -> |^|c =/= (/)))
138	102, 137	syld 30	. . . . . . . . . 10 |- (((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) /\ c =/= (/)) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/)))
139	138	ex 402	. . . . . . . . 9 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> (c =/= (/) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/))))
140	60, 139	pm2.61dne 2091	. . . . . . . 8 |- ((J e. Top /\ c C_ (Clsd` J) /\ (/) e/ ( fi ` c)) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/)))
141	140	3exp 1066	. . . . . . 7 |- (J e. Top -> (c C_ (Clsd` J) -> ((/) e/ ( fi ` c) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/)))))
142	72	elpw 3037	. . . . . . 7 |- (c e. ~P(Clsd` J) <-> c C_ (Clsd` J))
143	141, 142	syl5ib 223	. . . . . 6 |- (J e. Top -> (c e. ~P(Clsd` J) -> ((/) e/ ( fi ` c) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/)))))
144	143	com23 36	. . . . 5 |- (J e. Top -> ((/) e/ ( fi ` c) -> (c e. ~P(Clsd` J) -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> |^|c =/= (/)))))
145	144	com24 41	. . . 4 |- (J e. Top -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> (c e. ~P(Clsd` J) -> ((/) e/ ( fi ` c) -> |^|c =/= (/)))))
146	145	r19.21adv 2181	. . 3 |- (J e. Top -> (A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/)) -> A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/))))
147	52, 146	impbid 574	. 2 |- (J e. Top -> (A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/)) <-> A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/))))
148		compfipin0 15436	. 2 |- (J e. Top -> (J e. Comp <-> A.c e. ~P (Clsd` J)((/) e/ ( fi ` c) -> |^|c =/= (/))))
149		eqid 1884	. . . . . . 7 |- U.f = U.f
150	3, 149	filclus 15605	. . . . . 6 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> ((fClus` J)` f) = |^|_s e. f ((cls` J)` s))
151	150	neeq1d 2028	. . . . 5 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (((fClus` J)` f) =/= (/) <-> |^|_s e. f ((cls` J)` s) =/= (/)))
152	151	3expa 1067	. . . 4 |- (((J e. Top /\ f e. Fil) /\ X = U.f) -> (((fClus` J)` f) =/= (/) <-> |^|_s e. f ((cls` J)` s) =/= (/)))
153	152	pm5.74da 646	. . 3 |- ((J e. Top /\ f e. Fil) -> ((X = U.f -> ((fClus` J)` f) =/= (/)) <-> (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/))))
154	153	ralbidva 2119	. 2 |- (J e. Top -> (A.f e. Fil (X = U.f -> ((fClus` J)` f) =/= (/)) <-> A.f e. Fil (X = U.f -> |^|_s e. f ((cls` J)` s) =/= (/))))
155	147, 148, 154	3bitr4d 609	1 |- (J e. Top -> (J e. Comp <-> A.f e. Fil (X = U.f -> ((fClus` J)` f) =/= (/))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017   e/ wnel 2018  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  U.cuni 3177  |^|cint 3214  |^|_ciin 3256  ` cfv 3998  Topctop 8857  Clsdccld 8936  clsccl 8938   fi cfi 10210  fBascfbas 10257  filGencfg 10258  Filcfil 10264  Compccomp 10328  fCluscfclus 15582

This theorem is referenced by:  ufcomp 15622

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-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-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-cls 8941  df-fi 10211  df-fbas 10259  df-fg 10260  df-fil 10265  df-comp 10329  df-fclus 15584

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