Theorem ufilcmp 21646

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 21518	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘∪ 𝐽) → 𝑓 ∈ (Fil‘∪ 𝐽))
2		eqid 2610	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclscmpi 21643	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
4	1, 3	sylan2 490	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
5	4	ralrimiva 2949	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6		toponuni 20542	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	fveq2d 6107	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘∪ 𝐽))
8	7	raleqdv 3121	. . . . 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	syl5ibr 235	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11		ufli 21528	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
12	11	adantlr 747	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
13		r19.29 3054	. . . . . . 7 ⊢ ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓))
14		simpllr 795	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15		simplr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16		simprr 792	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ⊆ 𝑓)
17		fclsss2 21637	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
18	14, 15, 16, 17	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19		ssn0 3928	. . . . . . . . . . . . 13 ⊢ (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
20	19	ex 449	. . . . . . . . . . . 12 ⊢ ((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
21	18, 20	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
22	21	expr 641	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔 ⊆ 𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
23	22	com23 84	. . . . . . . . 9 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝑔 ⊆ 𝑓 → (𝐽 fClus 𝑔) ≠ ∅)))
24	23	impd 446	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
25	24	rexlimdva 3013	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
26	13, 25	syl5 33	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
27	12, 26	mpan2d 706	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
28	27	ralrimdva 2952	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
29		fclscmp 21644	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
30	29	adantl 481	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
31	28, 30	sylibrd 248	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
32	10, 31	impbid 201	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
33		uffclsflim 21645	. . . 4 ⊢ (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
34	33	neeq1d 2841	. . 3 ⊢ (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
35	34	ralbiia 2962	. 2 ⊢ (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
36	32, 35	syl6bb 275	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549  TopOnctopon 20518  Compccmp 20999  Filcfil 21459  UFilcufil 21513  UFLcufl 21514   fLim cflim 21548   fClus cfcls 21550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-fbas 19564  df-fg 19565  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cmp 21000  df-fil 21460  df-ufil 21515  df-ufl 21516  df-flim 21553  df-fcls 21555

This theorem is referenced by:  alexsub  21659