Theorem fclscmpi 23753

Description: Forward direction of fclscmp 23754. 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 23119	. . . 4 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		flimfnfcls.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	fclsval 23732	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅))
4		eqid 2730	. . . . . 6 ⊢ 𝑋 = 𝑋
5	4	iftruei 4534	. . . . 5 ⊢ if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥)
6	3, 5	eqtrdi 2786	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
7	1, 6	sylan 578	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
8		fvex 6903	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
9	8	dfiin3 5965	. . 3 ⊢ ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))
10	7, 9	eqtrdi 2786	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11		simpl 481	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
12	11	adantr 479	. . . . . . 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 23576	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
15	14	adantll 710	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
16	2	clscld 22771	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
17	13, 15, 16	syl2anc 582	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18	17	fmpttd 7115	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
19	18	frnd 6724	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20		simpr 483	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
21	20	adantr 479	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
22		simpr 483	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
23	2	clsss3 22783	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
24	13, 15, 23	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
25	2	sscls 22780	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
26	13, 15, 25	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27		filss 23577	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
29	28	fmpttd 7115	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶𝐹)
30	29	frnd 6724	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31		fiss 9421	. . . . . 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 582	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33		filfi 23583	. . . . . 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 4021	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36		0nelfil 23573	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
37	20, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
38	35, 37	ssneldd 3984	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39		cmpfii 23133	. . 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 1369	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
41	10, 40	eqnetrd 3006	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938   ⊆ wss 3947  ∅c0 4321  ifcif 4527  ∪ cuni 4907  ∩ cint 4949  ∩ ciin 4997   ↦ cmpt 5230  ran crn 5676  ‘cfv 6542  (class class class)co 7411  ficfi 9407  Topctop 22615  Clsdccld 22740  clsccl 22742  Compccmp 23110  Filcfil 23569   fClus cfcls 23660

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 21141  df-top 22616  df-cld 22743  df-cls 22745  df-cmp 23111  df-fil 23570  df-fcls 23665

This theorem is referenced by:  fclscmp  23754  ufilcmp  23756  relcmpcmet  25066