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

Theorem locfincmp 22423
Description: For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
locfincmp.1 𝑋 = 𝐽
locfincmp.2 𝑌 = 𝐶
Assertion
Ref Expression
locfincmp (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Proof of Theorem locfincmp
Dummy variables 𝑜 𝑐 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfincmp.1 . . . . . . . . . 10 𝑋 = 𝐽
21locfinnei 22420 . . . . . . . . 9 ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥𝑋) → ∃𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
32ralrimiva 3105 . . . . . . . 8 (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41cmpcov2 22287 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
53, 4sylan2 596 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
6 elfpw 8978 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐𝐽𝑐 ∈ Fin))
7 simplrr 778 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → 𝑐 ∈ Fin)
8 eldifsn 4700 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥𝐶𝑥 ≠ ∅))
9 ineq1 4120 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → (𝑠𝑜) = (𝑥𝑜))
109neeq1d 3000 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑥 → ((𝑠𝑜) ≠ ∅ ↔ (𝑥𝑜) ≠ ∅))
11 simplrl 777 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥𝐶)
12 simplrr 778 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑥)
13 simprr 773 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑜)
14 inelcm 4379 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑦𝑜) → (𝑥𝑜) ≠ ∅)
1512, 13, 14syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → (𝑥𝑜) ≠ ∅)
1610, 11, 15elrabd 3604 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
17 elunii 4824 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑌 = 𝐶
1917, 18eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝑥𝑥𝐶) → 𝑦𝑌)
2019ancoms 462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐶𝑦𝑥) → 𝑦𝑌)
2120adantl 485 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑌)
221, 18locfinbas 22419 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
2322adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
2423ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑌)
2521, 24eleqtrrd 2841 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑋)
26 simplr 769 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑐)
2725, 26eleqtrd 2840 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦 𝑐)
28 eluni2 4823 . . . . . . . . . . . . . . . . . . 19 (𝑦 𝑐 ↔ ∃𝑜𝑐 𝑦𝑜)
2927, 28sylib 221 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑦𝑜)
3016, 29reximddv 3194 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3130expr 460 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3231exlimdv 1941 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (∃𝑦 𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
33 n0 4261 . . . . . . . . . . . . . . 15 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
34 eliun 4908 . . . . . . . . . . . . . . 15 (𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ↔ ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3532, 33, 343imtr4g 299 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑥 ≠ ∅ → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3635expimpd 457 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → ((𝑥𝐶𝑥 ≠ ∅) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
378, 36syl5bi 245 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3837ssrdv 3907 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
39 iunfi 8964 . . . . . . . . . . . . 13 ((𝑐 ∈ Fin ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)
4039ex 416 . . . . . . . . . . . 12 (𝑐 ∈ Fin → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41 ssfi 8851 . . . . . . . . . . . . 13 (( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
4241expcom 417 . . . . . . . . . . . 12 ((𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} → ( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4340, 42sylan9 511 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
447, 38, 43syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4544expimpd 457 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
466, 45sylan2b 597 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
4746rexlimdva 3203 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
485, 47mpd 15 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∖ {∅}) ∈ Fin)
49 snfi 8721 . . . . . 6 {∅} ∈ Fin
50 unfi 8850 . . . . . 6 (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
5148, 49, 50sylancl 589 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52 ssun1 4086 . . . . . 6 𝐶 ⊆ (𝐶 ∪ {∅})
53 undif1 4390 . . . . . 6 ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
5452, 53sseqtrri 3938 . . . . 5 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55 ssfi 8851 . . . . 5 ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
5651, 54, 55sylancl 589 . . . 4 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
5756, 23jca 515 . . 3 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
5857ex 416 . 2 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59 cmptop 22292 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
601, 18finlocfin 22417 . . . 4 ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61603expib 1124 . . 3 (𝐽 ∈ Top → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
6259, 61syl 17 . 2 (𝐽 ∈ Comp → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
6358, 62impbid 215 1 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wne 2940  wral 3061  wrex 3062  {crab 3065  cdif 3863  cun 3864  cin 3865  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541   cuni 4819   ciun 4904  cfv 6380  Fincfn 8626  Topctop 21790  Compccmp 22283  LocFinclocfin 22401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-1o 8202  df-en 8627  df-fin 8630  df-top 21791  df-cmp 22284  df-locfin 22404
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator