MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufilcmp Structured version   Visualization version   GIF version

Theorem ufilcmp 23383
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 23255 . . . . . 6 (𝑓 ∈ (UFil‘ 𝐽) → 𝑓 ∈ (Fil‘ 𝐽))
2 eqid 2736 . . . . . . 7 𝐽 = 𝐽
32fclscmpi 23380 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
41, 3sylan2 593 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑓 ∈ (UFil‘ 𝐽)) → (𝐽 fClus 𝑓) ≠ ∅)
54ralrimiva 3143 . . . 4 (𝐽 ∈ Comp → ∀𝑓 ∈ (UFil‘ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)
6 toponuni 22263 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
76fveq2d 6846 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → (UFil‘𝑋) = (UFil‘ 𝐽))
87raleqdv 3313 . . . . 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 23265 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
1211adantlr 713 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓)
13 r19.29 3117 . . . . . . 7 ((∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ∧ ∃𝑓 ∈ (UFil‘𝑋)𝑔𝑓) → ∃𝑓 ∈ (UFil‘𝑋)((𝐽 fClus 𝑓) ≠ ∅ ∧ 𝑔𝑓))
14 simpllr 774 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
15 simplr 767 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔 ∈ (Fil‘𝑋))
16 simprr 771 . . . . . . . . . . . 12 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → 𝑔𝑓)
17 fclsss2 23374 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝑔𝑓) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
1814, 15, 16, 17syl3anc 1371 . . . . . . . . . . 11 ((((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ (𝑓 ∈ (UFil‘𝑋) ∧ 𝑔𝑓)) → (𝐽 fClus 𝑓) ⊆ (𝐽 fClus 𝑔))
19 ssn0 4360 . . . . . . . . . . . 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 3152 . . . . . . 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 692 . . . . 5 (((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑔 ∈ (Fil‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus 𝑔) ≠ ∅))
2726ralrimdva 3151 . . . 4 ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∀𝑓 ∈ (UFil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑔 ∈ (Fil‘𝑋)(𝐽 fClus 𝑔) ≠ ∅))
28 fclscmp 23381 . . . . 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 23382 . . . 4 (𝑓 ∈ (UFil‘𝑋) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
3332neeq1d 3003 . . 3 (𝑓 ∈ (UFil‘𝑋) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
3433ralbiia 3094 . 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 2106  wne 2943  wral 3064  wrex 3073  wss 3910  c0 4282   cuni 4865  cfv 6496  (class class class)co 7357  TopOnctopon 22259  Compccmp 22737  Filcfil 23196  UFilcufil 23250  UFLcufl 23251   fLim cflim 23285   fClus cfcls 23287
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-fin 8887  df-fi 9347  df-fbas 20793  df-fg 20794  df-top 22243  df-topon 22260  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-cmp 22738  df-fil 23197  df-ufil 23252  df-ufl 23253  df-flim 23290  df-fcls 23292
This theorem is referenced by:  alexsub  23396
  Copyright terms: Public domain W3C validator