Theorem fclscmpi 23977

Description: Forward direction of fclscmp 23978. 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 23343	. . . 4 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		flimfnfcls.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	fclsval 23956	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅))
4		eqid 2737	. . . . . 6 ⊢ 𝑋 = 𝑋
5	4	iftruei 4487	. . . . 5 ⊢ if(𝑋 = 𝑋, ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥), ∅) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥)
6	3, 5	eqtrdi 2788	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
7	1, 6	sylan 581	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥))
8		fvex 6848	. . . 4 ⊢ ((cls‘𝐽)‘𝑥) ∈ V
9	8	dfiin3 5921	. . 3 ⊢ ∩ 𝑥 ∈ 𝐹 ((cls‘𝐽)‘𝑥) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))
10	7, 9	eqtrdi 2788	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11		simpl 482	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
12	11	adantr 480	. . . . . . 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 23800	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
15	14	adantll 715	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝑋)
16	2	clscld 22995	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
17	13, 15, 16	syl2anc 585	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18	17	fmpttd 7062	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
19	18	frnd 6671	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20		simpr 484	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
21	20	adantr 480	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
22		simpr 484	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ 𝐹)
23	2	clsss3 23007	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
24	13, 15, 23	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
25	2	sscls 23004	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
26	13, 15, 25	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27		filss 23801	. . . . . . . . 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 1375	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
29	28	fmpttd 7062	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶𝐹)
30	29	frnd 6671	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31		fiss 9331	. . . . . 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 585	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33		filfi 23807	. . . . . 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 3971	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36		0nelfil 23797	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
37	20, 36	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
38	35, 37	ssneldd 3937	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39		cmpfii 23357	. . 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 1374	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ∩ ran (𝑥 ∈ 𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
41	10, 40	eqnetrd 3000	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ⊆ wss 3902  ∅c0 4286  ifcif 4480  ∪ cuni 4864  ∩ cint 4903  ∩ ciin 4948   ↦ cmpt 5180  ran crn 5626  ‘cfv 6493  (class class class)co 7360  ficfi 9317  Topctop 22841  Clsdccld 22964  clsccl 22966  Compccmp 23334  Filcfil 23793   fClus cfcls 23884

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1o 8399  df-2o 8400  df-en 8888  df-dom 8889  df-fin 8891  df-fi 9318  df-fbas 21310  df-top 22842  df-cld 22967  df-cls 22969  df-cmp 23335  df-fil 23794  df-fcls 23889

This theorem is referenced by:  fclscmp  23978  ufilcmp  23980  relcmpcmet  25278