Theorem ufilcmp 23266

Description: A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ufilcmp	⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))

Distinct variable groups:   𝑓,𝐽   𝑓,𝑋

Proof of Theorem ufilcmp

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ufilfil 23138	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘∪ 𝐽) → 𝑓 ∈ (Fil‘∪ 𝐽))
2		eqid 2737	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclscmpi 23263	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
4	1, 3	sylan2 593	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
5	4	ralrimiva 3140	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6		toponuni 22146	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	fveq2d 6816	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘∪ 𝐽))
8	7	raleqdv 3310	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
9	8	adantl 482	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
10	5, 9	syl5ibr 245	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11		ufli 23148	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
12	11	adantlr 712	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
13		r19.29 3114	. . . . . . 7 ⊢ ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓))
14		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15		simplr 766	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16		simprr 770	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ⊆ 𝑓)
17		fclsss2 23257	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
18	14, 15, 16, 17	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19		ssn0 4345	. . . . . . . . . . . 12 ⊢ (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
20	19	ex 413	. . . . . . . . . . 11 ⊢ ((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
21	18, 20	syl 17	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
22	21	expr 457	. . . . . . . . 9 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔 ⊆ 𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
23	22	impcomd 412	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
24	23	rexlimdva 3149	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
25	13, 24	syl5 34	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
26	12, 25	mpan2d 691	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
27	26	ralrimdva 3148	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28		fclscmp 23264	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
29	28	adantl 482	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
30	27, 29	sylibrd 258	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
31	10, 30	impbid 211	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
32		uffclsflim 23265	. . . 4 ⊢ (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
33	32	neeq1d 3001	. . 3 ⊢ (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
34	33	ralbiia 3091	. 2 ⊢ (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
35	31, 34	bitrdi 286	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2105   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ⊆ wss 3897  ∅c0 4267  ∪ cuni 4850  ‘cfv 6466  (class class class)co 7317  TopOnctopon 22142  Compccmp 22620  Filcfil 23079  UFilcufil 23133  UFLcufl 23134   fLim cflim 23168   fClus cfcls 23170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1o 8346  df-er 8548  df-en 8784  df-fin 8787  df-fi 9247  df-fbas 20677  df-fg 20678  df-top 22126  df-topon 22143  df-cld 22253  df-ntr 22254  df-cls 22255  df-nei 22332  df-cmp 22621  df-fil 23080  df-ufil 23135  df-ufl 23136  df-flim 23173  df-fcls 23175

This theorem is referenced by:  alexsub  23279