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Theorem ufilcmp 23266
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.
StepHypRef Expression
1 ufilfil 23138 . . . . . 6 (𝑓 ∈ (UFil‘ 𝐽) → 𝑓 ∈ (Fil‘ 𝐽))
2 eqid 2737 . . . . . . 7 𝐽 = 𝐽
32fclscmpi 23263 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
41, 3sylan2 593 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
54ralrimiva 3140 . . . 4 (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6 toponuni 22146 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76fveq2d 6816 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘ 𝐽))
87raleqdv 3310 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
98adantl 482 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
105, 9syl5ibr 245 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11 ufli 23148 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
1211adantlr 712 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
13 r19.29 3114 . . . . . . 7 ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓))
14 simpllr 773 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15 simplr 766 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16 simprr 770 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔𝑓)
17 fclsss2 23257 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
1814, 15, 16, 17syl3anc 1370 . . . . . . . . . . 11 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19 ssn0 4345 . . . . . . . . . . . 12 (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
2019ex 413 . . . . . . . . . . 11 ((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
2118, 20syl 17 . . . . . . . . . 10 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
2221expr 457 . . . . . . . . 9 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
2322impcomd 412 . . . . . . . 8 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
2423rexlimdva 3149 . . . . . . 7 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
2513, 24syl5 34 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
2612, 25mpan2d 691 . . . . 5 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
2726ralrimdva 3148 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28 fclscmp 23264 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
2928adantl 482 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
3027, 29sylibrd 258 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
3110, 30impbid 211 . 2 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
32 uffclsflim 23265 . . . 4 (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
3332neeq1d 3001 . . 3 (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
3433ralbiia 3091 . 2 (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
3531, 34bitrdi 286 1 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2105  wne 2941  wral 3062  wrex 3071  wss 3897  c0 4267   cuni 4850  cfv 6466  (class class class)co 7317  TopOnctopon 22142  Compccmp 22620  Filcfil 23079  UFilcufil 23133  UFLcufl 23134   fLim cflim 23168   fClus cfcls 23170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1o 8346  df-er 8548  df-en 8784  df-fin 8787  df-fi 9247  df-fbas 20677  df-fg 20678  df-top 22126  df-topon 22143  df-cld 22253  df-ntr 22254  df-cls 22255  df-nei 22332  df-cmp 22621  df-fil 23080  df-ufil 23135  df-ufl 23136  df-flim 23173  df-fcls 23175
This theorem is referenced by:  alexsub  23279
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