Theorem ufilcmp 24093

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 23965	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘∪ 𝐽) → 𝑓 ∈ (Fil‘∪ 𝐽))
2		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclscmpi 24090	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
4	1, 3	sylan2 602	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘∪ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
5	4	ralrimiva 3155	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6		toponuni 22975	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	fveq2d 6872	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘∪ 𝐽))
8	7	raleqdv 3321	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
9	8	adantl 485	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
10	5, 9	imbitrrid 248	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11		ufli 23975	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
12	11	adantlr 725	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓)
13		r19.29 3126	. . . . . . 7 ⊢ ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔 ⊆ 𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓))
14		simpllr 785	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15		simplr 778	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16		simprr 782	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → 𝑔 ⊆ 𝑓)
17		fclsss2 24084	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
18	14, 15, 16, 17	syl3anc 1391	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔 ⊆ 𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19		ssn0 4359	. . . . . . . . . . . 12 ⊢ (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
20	19	ex 416	. . . . . . . . . . 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 460	. . . . . . . . 9 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔 ⊆ 𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
23	22	impcomd 415	. . . . . . . 8 ⊢ ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔 ⊆ 𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
24	23	rexlimdva 3164	. . . . . . 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 704	. . . . 5 ⊢ (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
27	26	ralrimdva 3163	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28		fclscmp 24091	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
29	28	adantl 485	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
30	27, 29	sylibrd 261	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
31	10, 30	impbid 214	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
32		uffclsflim 24092	. . . 4 ⊢ (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
33	32	neeq1d 3017	. . 3 ⊢ (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
34	33	ralbiia 3107	. 2 ⊢ (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
35	31, 34	bitrdi 289	1 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  ∃wrex 3087   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4866  ‘cfv 6522  (class class class)co 7397  TopOnctopon 22971  Compccmp 23447  Filcfil 23906  UFilcufil 23960  UFLcufl 23961   fLim cflim 23995   fClus cfcls 23997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1o 8438  df-2o 8439  df-en 8929  df-dom 8930  df-fin 8932  df-fi 9358  df-fbas 21422  df-fg 21423  df-top 22955  df-topon 22972  df-cld 23080  df-ntr 23081  df-cls 23082  df-nei 23159  df-cmp 23448  df-fil 23907  df-ufil 23962  df-ufl 23963  df-flim 24000  df-fcls 24002

This theorem is referenced by:  alexsub  24106