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Theorem ufilcmp 22344
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 22216 . . . . . 6 (𝑓 ∈ (UFil‘ 𝐽) → 𝑓 ∈ (Fil‘ 𝐽))
2 eqid 2778 . . . . . . 7 𝐽 = 𝐽
32fclscmpi 22341 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
41, 3sylan2 583 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
54ralrimiva 3132 . . . 4 (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6 toponuni 21226 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76fveq2d 6503 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘ 𝐽))
87raleqdv 3355 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
98adantl 474 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅))
105, 9syl5ibr 238 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
11 ufli 22226 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
1211adantlr 702 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
13 r19.29 3200 . . . . . . 7 ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓))
14 simpllr 763 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15 simplr 756 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16 simprr 760 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔𝑓)
17 fclsss2 22335 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
1814, 15, 16, 17syl3anc 1351 . . . . . . . . . . 11 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19 ssn0 4240 . . . . . . . . . . . 12 (((𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔) ∧ (𝐽 fClus 𝑓) ≠ ∅) → (𝐽 fClus 𝑔) ≠ ∅)
2019ex 405 . . . . . . . . . . 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 449 . . . . . . . . 9 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝑔𝑓 → ((𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅)))
2322impcomd 403 . . . . . . . 8 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝑓 ∈ (UFil‘𝑋)) → (((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓) → (𝐽 fClus 𝑔) ≠ ∅))
2423rexlimdva 3229 . . . . . . 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 681 . . . . 5 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
2726ralrimdva 3139 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28 fclscmp 22342 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
2928adantl 474 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
3027, 29sylibrd 251 . . 3 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp))
3110, 30impbid 204 . 2 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
32 uffclsflim 22343 . . . 4 (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
3332neeq1d 3026 . . 3 (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
3433ralbiia 3114 . 2 (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)
3531, 34syl6bb 279 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 198  wa 387  wcel 2050  wne 2967  wral 3088  wrex 3089  wss 3829  c0 4178   cuni 4712  cfv 6188  (class class class)co 6976  TopOnctopon 21222  Compccmp 21698  Filcfil 22157  UFilcufil 22211  UFLcufl 22212   fLim cflim 22246   fClus cfcls 22248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-2o 7906  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fi 8670  df-fbas 20244  df-fg 20245  df-top 21206  df-topon 21223  df-cld 21331  df-ntr 21332  df-cls 21333  df-nei 21410  df-cmp 21699  df-fil 22158  df-ufil 22213  df-ufl 22214  df-flim 22251  df-fcls 22253
This theorem is referenced by:  alexsub  22357
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