Theorem fclscmp 24262

Description: A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
fclscmp	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))

Distinct variable groups:   𝑓,𝐽   𝑓,𝑋

Proof of Theorem fclscmp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclscmpi 24261	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
3	2	ralrimiva 3156	. . 3 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
4		toponuni 23145	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
5	4	fveq2d 6886	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽))
6	5	raleqdv 3321	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
7	3, 6	imbitrrid 249	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
8		elpwi 4567	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
9		vn0 4294	. . . . . . . . . 10 ⊢ V ≠ ∅
10		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
11	10	inteqd 4915	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 = ∩ ∅)
12		int0 4925	. . . . . . . . . . . 12 ⊢ ∩ ∅ = V
13	11, 12	eqtrdi 2813	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 = V)
14	13	neeq1d 3016	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∩ 𝑥 ≠ ∅ ↔ V ≠ ∅))
15	9, 14	mpbiri 261	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 ≠ ∅)
16	15	a1d 26	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
17		ssfii 9393	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥))
18	17	elv 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ⊆ (fi‘𝑥)
19		simplrl 789	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽))
20	1	cldss2 23261	. . . . . . . . . . . . . . . . . . 19 ⊢ (Clsd‘𝐽) ⊆ 𝒫 ∪ 𝐽
21	4	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = ∪ 𝐽)
22	21	pweqd 4577	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
23	20, 22	sseqtrrid 3977	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
24	19, 23	sstrd 3944	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑋)
25		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
26		simplrr 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈ (fi‘𝑥))
27		toponmax 23157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
28	27	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 ∈ 𝐽)
29		fsubbas 24099	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐽 → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋 ∧ 𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
30	28, 29	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋 ∧ 𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
31	24, 25, 26, 30	mpbir3and 1361	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋))
32		ssfg 24104	. . . . . . . . . . . . . . . 16 ⊢ ((fi‘𝑥) ∈ (fBas‘𝑋) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
33	31, 32	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
34	18, 33	sstrid 3945	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥)))
35	34	sselda 3934	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥)))
36		fclssscls 24250	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑋filGen(fi‘𝑥)) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
37	35, 36	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
38	19	sselda 3934	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (Clsd‘𝐽))
39		cldcls 23273	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑦) = 𝑦)
40	38, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → ((cls‘𝐽)‘𝑦) = 𝑦)
41	37, 40	sseqtrd 3970	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
42	41	ralrimiva 3156	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
43		ssint 4927	. . . . . . . . . 10 ⊢ ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥 ↔ ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
44	42, 43	sylibr 237	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥)
45		fgcl 24110	. . . . . . . . . 10 ⊢ ((fi‘𝑥) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋))
46		oveq2 7425	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥))))
47	46	neeq1d 3016	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
48	47	rspcv 3575	. . . . . . . . . 10 ⊢ ((𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
49	31, 45, 48	3syl 19	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
50		ssn0 4358	. . . . . . . . 9 ⊢ (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → ∩ 𝑥 ≠ ∅)
51	44, 49, 50	syl6an 697	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
52	16, 51	pm2.61dane 3044	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
53	52	expr 462	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅)))
54	8, 53	sylan2 605	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅)))
55	54	com23 87	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
56	55	ralrimdva 3164	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
57		topontop 23144	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
58		cmpfi 23639	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
59	57, 58	syl 18	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
60	56, 59	sylibrd 262	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
61	7, 60	impbid 215	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  ‘cfv 6537  (class class class)co 7417  ficfi 9384  fBascfbas 21579  filGencfg 21580  Topctop 23124  TopOnctopon 23141  Clsdccld 23247  clsccl 23249  Compccmp 23617  Filcfil 24077   fClus cfcls 24168

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1o 8459  df-2o 8460  df-en 8957  df-dom 8958  df-fin 8960  df-fi 9385  df-fbas 21588  df-fg 21589  df-top 23125  df-topon 23142  df-cld 23250  df-cls 23252  df-cmp 23618  df-fil 24078  df-fcls 24173

This theorem is used by:  ufilcmp  24264