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Theorem ufilcmp 23756
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 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
Distinct variable groups:   𝑓,𝐽   𝑓,𝑋

Proof of Theorem ufilcmp
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 ufilfil 23628 . . . . . 6 (𝑓 ∈ (UFilβ€˜βˆͺ 𝐽) β†’ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽))
2 eqid 2732 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
32fclscmpi 23753 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
41, 3sylan2 593 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFilβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
54ralrimiva 3146 . . . 4 (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (UFilβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…)
6 toponuni 22636 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76fveq2d 6895 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (UFilβ€˜π‘‹) = (UFilβ€˜βˆͺ 𝐽))
87raleqdv 3325 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ↔ βˆ€π‘“ ∈ (UFilβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…))
98adantl 482 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ↔ βˆ€π‘“ ∈ (UFilβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…))
105, 9imbitrrid 245 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
11 ufli 23638 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)𝑔 βŠ† 𝑓)
1211adantlr 713 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)𝑔 βŠ† 𝑓)
13 r19.29 3114 . . . . . . 7 ((βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ∧ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)𝑔 βŠ† 𝑓) β†’ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)((𝐽 fClus 𝑓) β‰  βˆ… ∧ 𝑔 βŠ† 𝑓))
14 simpllr 774 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 simplr 767 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
16 simprr 771 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝑔 βŠ† 𝑓)
17 fclsss2 23747 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔))
1814, 15, 16, 17syl3anc 1371 . . . . . . . . . . 11 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ (𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔))
19 ssn0 4400 . . . . . . . . . . . 12 (((𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) β‰  βˆ…) β†’ (𝐽 fClus 𝑔) β‰  βˆ…)
2019ex 413 . . . . . . . . . . 11 ((𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔) β†’ ((𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2118, 20syl 17 . . . . . . . . . 10 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ ((𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2221expr 457 . . . . . . . . 9 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ (𝑔 βŠ† 𝑓 β†’ ((𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus 𝑔) β‰  βˆ…)))
2322impcomd 412 . . . . . . . 8 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ 𝑓 ∈ (UFilβ€˜π‘‹)) β†’ (((𝐽 fClus 𝑓) β‰  βˆ… ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2423rexlimdva 3155 . . . . . . 7 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ (βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)((𝐽 fClus 𝑓) β‰  βˆ… ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2513, 24syl5 34 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ ((βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ∧ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)𝑔 βŠ† 𝑓) β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2612, 25mpan2d 692 . . . . 5 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ (𝐽 fClus 𝑔) β‰  βˆ…))
2726ralrimdva 3154 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑔) β‰  βˆ…))
28 fclscmp 23754 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑔) β‰  βˆ…))
2928adantl 482 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑔) β‰  βˆ…))
3027, 29sylibrd 258 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… β†’ 𝐽 ∈ Comp))
3110, 30impbid 211 . 2 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
32 uffclsflim 23755 . . . 4 (𝑓 ∈ (UFilβ€˜π‘‹) β†’ (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
3332neeq1d 3000 . . 3 (𝑓 ∈ (UFilβ€˜π‘‹) β†’ ((𝐽 fClus 𝑓) β‰  βˆ… ↔ (𝐽 fLim 𝑓) β‰  βˆ…))
3433ralbiia 3091 . 2 (βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ… ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…)
3531, 34bitrdi 286 1 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fLim 𝑓) β‰  βˆ…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  TopOnctopon 22632  Compccmp 23110  Filcfil 23569  UFilcufil 23623  UFLcufl 23624   fLim cflim 23658   fClus cfcls 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cmp 23111  df-fil 23570  df-ufil 23625  df-ufl 23626  df-flim 23663  df-fcls 23665
This theorem is referenced by:  alexsub  23769
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