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Theorem ufilcmp 23406
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 23278 . . . . . 6 (𝑓 ∈ (UFilβ€˜βˆͺ 𝐽) β†’ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽))
2 eqid 2733 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
32fclscmpi 23403 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
41, 3sylan2 594 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFilβ€˜βˆͺ 𝐽)) β†’ (𝐽 fClus 𝑓) β‰  βˆ…)
54ralrimiva 3140 . . . 4 (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (UFilβ€˜βˆͺ 𝐽)(𝐽 fClus 𝑓) β‰  βˆ…)
6 toponuni 22286 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
76fveq2d 6850 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (UFilβ€˜π‘‹) = (UFilβ€˜βˆͺ 𝐽))
87raleqdv 3312 . . . . 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 483 . . . 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, 9syl5ibr 246 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp β†’ βˆ€π‘“ ∈ (UFilβ€˜π‘‹)(𝐽 fClus 𝑓) β‰  βˆ…))
11 ufli 23288 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Filβ€˜π‘‹)) β†’ βˆƒπ‘“ ∈ (UFilβ€˜π‘‹)𝑔 βŠ† 𝑓)
1211adantlr 714 . . . . . 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 775 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 simplr 768 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝑔 ∈ (Filβ€˜π‘‹))
16 simprr 772 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) ∧ 𝑔 ∈ (Filβ€˜π‘‹)) ∧ (𝑓 ∈ (UFilβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓)) β†’ 𝑔 βŠ† 𝑓)
17 fclsss2 23397 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑔 ∈ (Filβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑓) β†’ (𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔))
1814, 15, 16, 17syl3anc 1372 . . . . . . . . . . 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 4364 . . . . . . . . . . . 12 (((𝐽 fClus 𝑓) βŠ† (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) β‰  βˆ…) β†’ (𝐽 fClus 𝑔) β‰  βˆ…)
2019ex 414 . . . . . . . . . . 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 458 . . . . . . . . 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 413 . . . . . . . 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 3149 . . . . . . 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 693 . . . . 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 3148 . . . 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 23404 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑔) β‰  βˆ…))
2928adantl 483 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐽 ∈ Comp ↔ βˆ€π‘” ∈ (Filβ€˜π‘‹)(𝐽 fClus 𝑔) β‰  βˆ…))
3027, 29sylibrd 259 . . 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 23405 . . . 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 287 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 397   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914  βˆ…c0 4286  βˆͺ cuni 4869  β€˜cfv 6500  (class class class)co 7361  TopOnctopon 22282  Compccmp 22760  Filcfil 23219  UFilcufil 23273  UFLcufl 23274   fLim cflim 23308   fClus cfcls 23310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1o 8416  df-er 8654  df-en 8890  df-fin 8893  df-fi 9355  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-cmp 22761  df-fil 23220  df-ufil 23275  df-ufl 23276  df-flim 23313  df-fcls 23315
This theorem is referenced by:  alexsub  23419
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