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

Theorem locfincmp 23022
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 23019 . . . . . . . . 9 ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥𝑋) → ∃𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
32ralrimiva 3147 . . . . . . . 8 (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41cmpcov2 22886 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
53, 4sylan2 594 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
6 elfpw 9351 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐𝐽𝑐 ∈ Fin))
7 simplrr 777 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → 𝑐 ∈ Fin)
8 eldifsn 4790 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥𝐶𝑥 ≠ ∅))
9 ineq1 4205 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → (𝑠𝑜) = (𝑥𝑜))
109neeq1d 3001 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑥 → ((𝑠𝑜) ≠ ∅ ↔ (𝑥𝑜) ≠ ∅))
11 simplrl 776 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥𝐶)
12 simplrr 777 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑥)
13 simprr 772 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑜)
14 inelcm 4464 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑦𝑜) → (𝑥𝑜) ≠ ∅)
1512, 13, 14syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → (𝑥𝑜) ≠ ∅)
1610, 11, 15elrabd 3685 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
17 elunii 4913 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑌 = 𝐶
1917, 18eleqtrrdi 2845 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝑥𝑥𝐶) → 𝑦𝑌)
2019ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐶𝑦𝑥) → 𝑦𝑌)
2120adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑌)
221, 18locfinbas 23018 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
2322adantl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
2423ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑌)
2521, 24eleqtrrd 2837 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑋)
26 simplr 768 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑐)
2725, 26eleqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦 𝑐)
28 eluni2 4912 . . . . . . . . . . . . . . . . . . 19 (𝑦 𝑐 ↔ ∃𝑜𝑐 𝑦𝑜)
2927, 28sylib 217 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑦𝑜)
3016, 29reximddv 3172 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3130expr 458 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3231exlimdv 1937 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (∃𝑦 𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
33 n0 4346 . . . . . . . . . . . . . . 15 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
34 eliun 5001 . . . . . . . . . . . . . . 15 (𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ↔ ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3532, 33, 343imtr4g 296 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑥 ≠ ∅ → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3635expimpd 455 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → ((𝑥𝐶𝑥 ≠ ∅) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
378, 36biimtrid 241 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3837ssrdv 3988 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
39 iunfi 9337 . . . . . . . . . . . . 13 ((𝑐 ∈ Fin ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)
4039ex 414 . . . . . . . . . . . 12 (𝑐 ∈ Fin → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41 ssfi 9170 . . . . . . . . . . . . 13 (( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
4241expcom 415 . . . . . . . . . . . 12 ((𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} → ( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4340, 42sylan9 509 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
447, 38, 43syl2anc 585 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4544expimpd 455 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
466, 45sylan2b 595 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
4746rexlimdva 3156 . . . . . . 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 9041 . . . . . 6 {∅} ∈ Fin
50 unfi 9169 . . . . . 6 (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
5148, 49, 50sylancl 587 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52 ssun1 4172 . . . . . 6 𝐶 ⊆ (𝐶 ∪ {∅})
53 undif1 4475 . . . . . 6 ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
5452, 53sseqtrri 4019 . . . . 5 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55 ssfi 9170 . . . . 5 ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
5651, 54, 55sylancl 587 . . . 4 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
5756, 23jca 513 . . 3 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
5857ex 414 . 2 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59 cmptop 22891 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
601, 18finlocfin 23016 . . . 4 ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61603expib 1123 . . 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 211 1 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   cuni 4908   ciun 4997  cfv 6541  Fincfn 8936  Topctop 22387  Compccmp 22882  LocFinclocfin 23000
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7853  df-1o 8463  df-en 8937  df-fin 8940  df-top 22388  df-cmp 22883  df-locfin 23003
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator