Theorem fclscmp 23968

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 2735	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	fclscmpi 23967	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
3	2	ralrimiva 3132	. . 3 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
4		toponuni 22852	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
5	4	fveq2d 6880	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽))
6	5	raleqdv 3305	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
7	3, 6	imbitrrid 246	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
8		elpwi 4582	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
9		vn0 4320	. . . . . . . . . 10 ⊢ V ≠ ∅
10		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅)
11	10	inteqd 4927	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 = ∩ ∅)
12		int0 4938	. . . . . . . . . . . 12 ⊢ ∩ ∅ = V
13	11, 12	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 = V)
14	13	neeq1d 2991	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∩ 𝑥 ≠ ∅ ↔ V ≠ ∅))
15	9, 14	mpbiri 258	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 ≠ ∅)
16	15	a1d 25	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
17		ssfii 9431	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥))
18	17	elv 3464	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ⊆ (fi‘𝑥)
19		simplrl 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽))
20	1	cldss2 22968	. . . . . . . . . . . . . . . . . . 19 ⊢ (Clsd‘𝐽) ⊆ 𝒫 ∪ 𝐽
21	4	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = ∪ 𝐽)
22	21	pweqd 4592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
23	20, 22	sseqtrrid 4002	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
24	19, 23	sstrd 3969	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑋)
25		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
26		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈ (fi‘𝑥))
27		toponmax 22864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
28	27	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 ∈ 𝐽)
29		fsubbas 23805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐽 → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋 ∧ 𝑥 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝑥))))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . 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 1343	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋))
32		ssfg 23810	. . . . . . . . . . . . . . . 16 ⊢ ((fi‘𝑥) ∈ (fBas‘𝑋) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
33	31, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥)))
34	18, 33	sstrid 3970	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥)))
35	34	sselda 3958	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥)))
36		fclssscls 23956	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑋filGen(fi‘𝑥)) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦))
38	19	sselda 3958	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (Clsd‘𝐽))
39		cldcls 22980	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑦) = 𝑦)
40	38, 39	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → ((cls‘𝐽)‘𝑦) = 𝑦)
41	37, 40	sseqtrd 3995	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
42	41	ralrimiva 3132	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
43		ssint 4940	. . . . . . . . . 10 ⊢ ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥 ↔ ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦)
44	42, 43	sylibr 234	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥)
45		fgcl 23816	. . . . . . . . . 10 ⊢ ((fi‘𝑥) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋))
46		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥))))
47	46	neeq1d 2991	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
48	47	rspcv 3597	. . . . . . . . . 10 ⊢ ((𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
49	31, 45, 48	3syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅))
50		ssn0 4379	. . . . . . . . 9 ⊢ (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩ 𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → ∩ 𝑥 ≠ ∅)
51	44, 49, 50	syl6an 684	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
52	16, 51	pm2.61dane 3019	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅))
53	52	expr 456	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅)))
54	8, 53	sylan2 593	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥 ≠ ∅)))
55	54	com23 86	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
56	55	ralrimdva 3140	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
57		topontop 22851	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
58		cmpfi 23346	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
59	57, 58	syl 17	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
60	56, 59	sylibrd 259	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
61	7, 60	impbid 212	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  ∪ cuni 4883  ∩ cint 4922  ‘cfv 6531  (class class class)co 7405  ficfi 9422  fBascfbas 21303  filGencfg 21304  Topctop 22831  TopOnctopon 22848  Clsdccld 22954  clsccl 22956  Compccmp 23324  Filcfil 23783   fClus cfcls 23874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1o 8480  df-2o 8481  df-en 8960  df-dom 8961  df-fin 8963  df-fi 9423  df-fbas 21312  df-fg 21313  df-top 22832  df-topon 22849  df-cld 22957  df-cls 22959  df-cmp 23325  df-fil 23784  df-fcls 23879

This theorem is referenced by:  ufilcmp  23970