Theorem fclscmpi 22880

Description: Forward direction of fclscmp 22881. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
flimfnfcls.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
fclscmpi	⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Proof of Theorem fclscmpi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmptop 22246	. . . 4 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		flimfnfcls.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	fclsval 22859	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅))
4		eqid 2736	. . . . . 6 ⊢ 𝑋 = 𝑋
5	4	iftruei 4432	. . . . 5 ⊢ if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥)
6	3, 5	eqtrdi 2787	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
7	1, 6	sylan 583	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
8		fvex 6708	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
9	8	dfiin3 5821	. . 3 ⊢ ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))
10	7, 9	eqtrdi 2787	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11		simpl 486	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
12	11	adantr 484	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐽 ∈ Comp)
13	12, 1	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐽 ∈ Top)
14		filelss 22703	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
15	14	adantll 714	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
16	2	clscld 21898	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
17	13, 15, 16	syl2anc 587	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18	17	fmpttd 6910	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
19	18	frnd 6531	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20		simpr 488	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
21	20	adantr 484	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
22		simpr 488	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
23	2	clsss3 21910	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
24	13, 15, 23	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
25	2	sscls 21907	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
26	13, 15, 25	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27		filss 22704	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑋 ∧ 𝑥 ⊆ ((cls‘𝐽)‘𝑥))) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
28	21, 22, 24, 26, 27	syl13anc 1374	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
29	28	fmpttd 6910	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶𝐹)
30	29	frnd 6531	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31		fiss 9018	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
32	20, 30, 31	syl2anc 587	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33		filfi 22710	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
34	20, 33	syl 17	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
35	32, 34	sseqtrd 3927	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36		0nelfil 22700	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
37	20, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
38	35, 37	ssneldd 3890	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39		cmpfii 22260	. . 3 ⊢ ((𝐽 ∈ Comp ∧ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
40	11, 19, 38, 39	syl3anc 1373	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
41	10, 40	eqnetrd 2999	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112   ≠ wne 2932   ⊆ wss 3853  ∅c0 4223  ifcif 4425  ∪ cuni 4805  ∩ cint 4845  ∩ ciin 4891   ↦ cmpt 5120  ran crn 5537  ‘cfv 6358  (class class class)co 7191  ficfi 9004  Topctop 21744  Clsdccld 21867  clsccl 21869  Compccmp 22237  Filcfil 22696   fClus cfcls 22787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-iin 4893  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1o 8180  df-er 8369  df-en 8605  df-fin 8608  df-fi 9005  df-fbas 20314  df-top 21745  df-cld 21870  df-cls 21872  df-cmp 22238  df-fil 22697  df-fcls 22792

This theorem is referenced by:  fclscmp  22881  ufilcmp  22883  relcmpcmet  24169