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Theorem locfincmp 22141
 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 22138 . . . . . . . . 9 ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥𝑋) → ∃𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
32ralrimiva 3149 . . . . . . . 8 (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41cmpcov2 22005 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
53, 4sylan2 595 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
6 elfpw 8813 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐𝐽𝑐 ∈ Fin))
7 simplrr 777 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → 𝑐 ∈ Fin)
8 eldifsn 4680 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥𝐶𝑥 ≠ ∅))
9 ineq1 4131 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → (𝑠𝑜) = (𝑥𝑜))
109neeq1d 3046 . . . . . . . . . . . . . . . . . . 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 4372 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑦𝑜) → (𝑥𝑜) ≠ ∅)
1512, 13, 14syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → (𝑥𝑜) ≠ ∅)
1610, 11, 15elrabd 3630 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
17 elunii 4806 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
18 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑌 = 𝐶
1917, 18eleqtrrdi 2901 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝑥𝑥𝐶) → 𝑦𝑌)
2019ancoms 462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐶𝑦𝑥) → 𝑦𝑌)
2120adantl 485 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑌)
221, 18locfinbas 22137 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
2322adantl 485 . . . . . . . . . . . . . . . . . . . . . 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 2893 . . . . . . . . . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦 𝑐)
28 eluni2 4805 . . . . . . . . . . . . . . . . . . 19 (𝑦 𝑐 ↔ ∃𝑜𝑐 𝑦𝑜)
2927, 28sylib 221 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑦𝑜)
3016, 29reximddv 3234 . . . . . . . . . . . . . . . . 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 1934 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (∃𝑦 𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
33 n0 4260 . . . . . . . . . . . . . . 15 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
34 eliun 4886 . . . . . . . . . . . . . . 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 3921 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
39 iunfi 8799 . . . . . . . . . . . . 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 8725 . . . . . . . . . . . . 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 596 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
4746rexlimdva 3243 . . . . . . 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 8580 . . . . . 6 {∅} ∈ Fin
50 unfi 8772 . . . . . 6 (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
5148, 49, 50sylancl 589 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52 ssun1 4099 . . . . . 6 𝐶 ⊆ (𝐶 ∪ {∅})
53 undif1 4382 . . . . . 6 ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
5452, 53sseqtrri 3952 . . . . 5 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55 ssfi 8725 . . . . 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 22010 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
601, 18finlocfin 22135 . . . 4 ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61603expib 1119 . . 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 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801  ∪ ciun 4882  ‘cfv 6325  Fincfn 8495  Topctop 21508  Compccmp 22001  LocFinclocfin 22119 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-fin 8499  df-top 21509  df-cmp 22002  df-locfin 22122 This theorem is referenced by: (None)
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