Theorem fclscmpi 22053

Description: Forward direction of fclscmp 22054. 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 21419	. . . 4 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		flimfnfcls.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	fclsval 22032	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅))
4		eqid 2771	. . . . . 6 ⊢ 𝑋 = 𝑋
5	4	iftruei 4233	. . . . 5 ⊢ if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥)
6	3, 5	syl6eq 2821	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
7	1, 6	sylan 569	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
8		fvex 6344	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
9	8	dfiin3 5518	. . 3 ⊢ ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))
10	7, 9	syl6eq 2821	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11		simpl 468	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
12	11	adantr 466	. . . . . . 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 21876	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
15	14	adantll 693	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
16	2	clscld 21072	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
17	13, 15, 16	syl2anc 573	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18	17	fmpttd 6530	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
19	18	frnd 6191	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20		simpr 471	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
21	20	adantr 466	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
22		simpr 471	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
23	2	clsss3 21084	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
24	13, 15, 23	syl2anc 573	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
25	2	sscls 21081	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
26	13, 15, 25	syl2anc 573	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27		filss 21877	. . . . . . . . 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 1478	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
29	28	fmpttd 6530	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶𝐹)
30	29	frnd 6191	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31		fiss 8490	. . . . . 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 573	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33		filfi 21883	. . . . . 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 3790	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36		0nelfil 21873	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
37	20, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
38	35, 37	ssneldd 3755	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39		cmpfii 21433	. . 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 1476	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
41	10, 40	eqnetrd 3010	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   ⊆ wss 3723  ∅c0 4063  ifcif 4226  ∪ cuni 4575  ∩ cint 4612  ∩ ciin 4656   ↦ cmpt 4864  ran crn 5251  ‘cfv 6030  (class class class)co 6796  ficfi 8476  Topctop 20918  Clsdccld 21041  clsccl 21043  Compccmp 21410  Filcfil 21869   fClus cfcls 21960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8477  df-fbas 19958  df-top 20919  df-cld 21044  df-cls 21046  df-cmp 21411  df-fil 21870  df-fcls 21965

This theorem is referenced by:  fclscmp  22054  ufilcmp  22056  relcmpcmet  23334