Theorem fclscmpi 23380

Description: Forward direction of fclscmp 23381. 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 22746	. . . 4 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		flimfnfcls.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	fclsval 23359	. . . . 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 4493	. . . . 5 ⊢ if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥)
6	3, 5	eqtrdi 2792	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
7	1, 6	sylan 580	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
8		fvex 6855	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
9	8	dfiin3 5922	. . 3 ⊢ ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))
10	7, 9	eqtrdi 2792	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11		simpl 483	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
12	11	adantr 481	. . . . . . 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 23203	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
15	14	adantll 712	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
16	2	clscld 22398	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
17	13, 15, 16	syl2anc 584	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18	17	fmpttd 7063	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
19	18	frnd 6676	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20		simpr 485	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
21	20	adantr 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
22		simpr 485	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
23	2	clsss3 22410	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
24	13, 15, 23	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
25	2	sscls 22407	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
26	13, 15, 25	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27		filss 23204	. . . . . . . . 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 1372	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
29	28	fmpttd 7063	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶𝐹)
30	29	frnd 6676	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31		fiss 9360	. . . . . 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 584	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33		filfi 23210	. . . . . 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 3984	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36		0nelfil 23200	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
37	20, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
38	35, 37	ssneldd 3947	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39		cmpfii 22760	. . 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 1371	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
41	10, 40	eqnetrd 3011	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   ⊆ wss 3910  ∅c0 4282  ifcif 4486  ∪ cuni 4865  ∩ cint 4907  ∩ ciin 4955   ↦ cmpt 5188  ran crn 5634  ‘cfv 6496  (class class class)co 7357  ficfi 9346  Topctop 22242  Clsdccld 22367  clsccl 22369  Compccmp 22737  Filcfil 23196   fClus cfcls 23287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-fin 8887  df-fi 9347  df-fbas 20793  df-top 22243  df-cld 22370  df-cls 22372  df-cmp 22738  df-fil 23197  df-fcls 23292

This theorem is referenced by:  fclscmp  23381  ufilcmp  23383  relcmpcmet  24682