Theorem ufilcmp 23974

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 23846	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘∪ 𝐽) → 𝑓 ∈ (Fil‘∪ 𝐽))
2		eqid 2734	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclscmpi 23971	. . . . . 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 3126	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6		toponuni 22856	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	fveq2d 6836	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘∪ 𝐽))
8	7	raleqdv 3294	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
9	8	adantl 481	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
10	5, 9	imbitrrid 246	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11		ufli 23856	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
12	11	adantlr 715	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
13		r19.29 3097	. . . . . . 7 ⊢ ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓))
14		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16		simprr 772	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ⊆ 𝑓)
17		fclsss2 23965	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
18	14, 15, 16, 17	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19		ssn0 4354	. . . . . . . . . . . 12 ⊢ (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
20	19	ex 412	. . . . . . . . . . 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 456	. . . . . . . . 9 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔 ⊆ 𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
23	22	impcomd 411	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
24	23	rexlimdva 3135	. . . . . . 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 694	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
27	26	ralrimdva 3134	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28		fclscmp 23972	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
29	28	adantl 481	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
30	27, 29	sylibrd 259	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
31	10, 30	impbid 212	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
32		uffclsflim 23973	. . . 4 ⊢ (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
33	32	neeq1d 2989	. . 3 ⊢ (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
34	33	ralbiia 3078	. 2 ⊢ (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
35	31, 34	bitrdi 287	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ∅c0 4283  ∪ cuni 4861  ‘cfv 6490  (class class class)co 7356  TopOnctopon 22852  Compccmp 23328  Filcfil 23787  UFilcufil 23841  UFLcufl 23842   fLim cflim 23876   fClus cfcls 23878

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1o 8395  df-2o 8396  df-en 8882  df-dom 8883  df-fin 8885  df-fi 9312  df-fbas 21304  df-fg 21305  df-top 22836  df-topon 22853  df-cld 22961  df-ntr 22962  df-cls 22963  df-nei 23040  df-cmp 23329  df-fil 23788  df-ufil 23843  df-ufl 23844  df-flim 23881  df-fcls 23883

This theorem is referenced by:  alexsub  23987