Theorem ufilcmp 23952

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 23824	. . . . . 6 ⊢ (𝑓 ∈ (UFil‘∪ 𝐽) → 𝑓 ∈ (Fil‘∪ 𝐽))
2		eqid 2729	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclscmpi 23949	. . . . . 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 3125	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6		toponuni 22834	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	fveq2d 6844	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘∪ 𝐽))
8	7	raleqdv 3296	. . . . 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 23834	. . . . . . 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 3094	. . . . . . 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 23943	. . . . . . . . . . . 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 4363	. . . . . . . . . . . 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 3134	. . . . . . 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 3133	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28		fclscmp 23950	. . . . 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 23951	. . . 4 ⊢ (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
33	32	neeq1d 2984	. . 3 ⊢ (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
34	33	ralbiia 3073	. 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 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4867  ‘cfv 6499  (class class class)co 7369  TopOnctopon 22830  Compccmp 23306  Filcfil 23765  UFilcufil 23819  UFLcufl 23820   fLim cflim 23854   fClus cfcls 23856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1o 8411  df-2o 8412  df-en 8896  df-dom 8897  df-fin 8899  df-fi 9338  df-fbas 21293  df-fg 21294  df-top 22814  df-topon 22831  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-cmp 23307  df-fil 23766  df-ufil 23821  df-ufl 23822  df-flim 23859  df-fcls 23861

This theorem is referenced by:  alexsub  23965