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Theorem locfincmp 23549
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 23546 . . . . . . . . 9 ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥𝑋) → ∃𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
32ralrimiva 3143 . . . . . . . 8 (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41cmpcov2 23413 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
53, 4sylan2 593 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
6 elfpw 9391 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐𝐽𝑐 ∈ Fin))
7 simplrr 778 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → 𝑐 ∈ Fin)
8 eldifsn 4790 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥𝐶𝑥 ≠ ∅))
9 ineq1 4220 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → (𝑠𝑜) = (𝑥𝑜))
109neeq1d 2997 . . . . . . . . . . . . . . . . . . 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 4470 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑦𝑜) → (𝑥𝑜) ≠ ∅)
1512, 13, 14syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → (𝑥𝑜) ≠ ∅)
1610, 11, 15elrabd 3696 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
17 elunii 4916 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑌 = 𝐶
1917, 18eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝑥𝑥𝐶) → 𝑦𝑌)
2019ancoms 458 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐶𝑦𝑥) → 𝑦𝑌)
2120adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑌)
221, 18locfinbas 23545 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
2322adantl 481 . . . . . . . . . . . . . . . . . . . . . 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 4915 . . . . . . . . . . . . . . . . . . 19 (𝑦 𝑐 ↔ ∃𝑜𝑐 𝑦𝑜)
2927, 28sylib 218 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑦𝑜)
3016, 29reximddv 3168 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3130expr 456 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3231exlimdv 1930 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (∃𝑦 𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
33 n0 4358 . . . . . . . . . . . . . . 15 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
34 eliun 4999 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → ((𝑥𝐶𝑥 ≠ ∅) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
378, 36biimtrid 242 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3837ssrdv 4000 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
39 iunfi 9380 . . . . . . . . . . . . 13 ((𝑐 ∈ Fin ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)
4039ex 412 . . . . . . . . . . . 12 (𝑐 ∈ Fin → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41 ssfi 9211 . . . . . . . . . . . . 13 (( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
4241expcom 413 . . . . . . . . . . . 12 ((𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} → ( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4340, 42sylan9 507 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
447, 38, 43syl2anc 584 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4544expimpd 453 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
466, 45sylan2b 594 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
4746rexlimdva 3152 . . . . . . 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 9081 . . . . . 6 {∅} ∈ Fin
50 unfi 9209 . . . . . 6 (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
5148, 49, 50sylancl 586 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52 ssun1 4187 . . . . . 6 𝐶 ⊆ (𝐶 ∪ {∅})
53 undif1 4481 . . . . . 6 ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
5452, 53sseqtrri 4032 . . . . 5 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55 ssfi 9211 . . . . 5 ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
5651, 54, 55sylancl 586 . . . 4 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
5756, 23jca 511 . . 3 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
5857ex 412 . 2 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59 cmptop 23418 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
601, 18finlocfin 23543 . . . 4 ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61603expib 1121 . . 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 212 1 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   cuni 4911   ciun 4995  cfv 6562  Fincfn 8983  Topctop 22914  Compccmp 23409  LocFinclocfin 23527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-en 8984  df-fin 8987  df-top 22915  df-cmp 23410  df-locfin 23530
This theorem is referenced by: (None)
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