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Theorem locfincmp 21551
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 21548 . . . . . . . . 9 ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥𝑋) → ∃𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
32ralrimiva 3115 . . . . . . . 8 (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
41cmpcov2 21415 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑜𝐽 (𝑥𝑜 ∧ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
53, 4sylan2 574 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
6 elfpw 8425 . . . . . . . . 9 (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐𝐽𝑐 ∈ Fin))
7 simplrr 757 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → 𝑐 ∈ Fin)
8 eldifsn 4454 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥𝐶𝑥 ≠ ∅))
9 simplrl 756 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥𝐶)
10 simplrr 757 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑥)
11 simprr 750 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑦𝑜)
12 inelcm 4176 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑦𝑜) → (𝑥𝑜) ≠ ∅)
1310, 11, 12syl2anc 567 . . . . . . . . . . . . . . . . . . 19 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → (𝑥𝑜) ≠ ∅)
14 ineq1 3959 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑥 → (𝑠𝑜) = (𝑥𝑜))
1514neeq1d 3002 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → ((𝑠𝑜) ≠ ∅ ↔ (𝑥𝑜) ≠ ∅))
1615elrab 3516 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ↔ (𝑥𝐶 ∧ (𝑥𝑜) ≠ ∅))
179, 13, 16sylanbrc 566 . . . . . . . . . . . . . . . . . 18 ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) ∧ (𝑜𝑐𝑦𝑜)) → 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
18 elunii 4580 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦𝑥𝑥𝐶) → 𝑦 𝐶)
19 locfincmp.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑌 = 𝐶
2018, 19syl6eleqr 2861 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦𝑥𝑥𝐶) → 𝑦𝑌)
2120ancoms 455 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝐶𝑦𝑥) → 𝑦𝑌)
2221adantl 467 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑌)
231, 19locfinbas 21547 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
2423adantl 467 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
2524ad3antrrr 703 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑌)
2622, 25eleqtrrd 2853 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦𝑋)
27 simplr 746 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑋 = 𝑐)
2826, 27eleqtrd 2852 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → 𝑦 𝑐)
29 eluni2 4579 . . . . . . . . . . . . . . . . . . 19 (𝑦 𝑐 ↔ ∃𝑜𝑐 𝑦𝑜)
3028, 29sylib 208 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑦𝑜)
3117, 30reximddv 3166 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ (𝑥𝐶𝑦𝑥)) → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3231expr 444 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3332exlimdv 2013 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (∃𝑦 𝑦𝑥 → ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
34 n0 4079 . . . . . . . . . . . . . . 15 (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦𝑥)
35 eliun 4659 . . . . . . . . . . . . . . 15 (𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ↔ ∃𝑜𝑐 𝑥 ∈ {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
3633, 34, 353imtr4g 285 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) ∧ 𝑥𝐶) → (𝑥 ≠ ∅ → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3736expimpd 441 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → ((𝑥𝐶𝑥 ≠ ∅) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
388, 37syl5bi 232 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}))
3938ssrdv 3759 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅})
40 iunfi 8411 . . . . . . . . . . . . 13 ((𝑐 ∈ Fin ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin)
4140ex 397 . . . . . . . . . . . 12 (𝑐 ∈ Fin → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin))
42 ssfi 8337 . . . . . . . . . . . . 13 (( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
4342expcom 398 . . . . . . . . . . . 12 ((𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} → ( 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4441, 43sylan9 493 . . . . . . . . . . 11 ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ 𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅}) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
457, 39, 44syl2anc 567 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) ∧ 𝑋 = 𝑐) → (∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
4645expimpd 441 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐𝐽𝑐 ∈ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
476, 46sylan2b 575 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
4847rexlimdva 3179 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑐 ∧ ∀𝑜𝑐 {𝑠𝐶 ∣ (𝑠𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
495, 48mpd 15 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∖ {∅}) ∈ Fin)
50 snfi 8195 . . . . . 6 {∅} ∈ Fin
51 unfi 8384 . . . . . 6 (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
5249, 50, 51sylancl 568 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
53 ssun1 3928 . . . . . 6 𝐶 ⊆ (𝐶 ∪ {∅})
54 undif1 4186 . . . . . 6 ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
5553, 54sseqtr4i 3788 . . . . 5 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
56 ssfi 8337 . . . . 5 ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
5752, 55, 56sylancl 568 . . . 4 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
5857, 24jca 497 . . 3 ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
5958ex 397 . 2 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
60 cmptop 21420 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
611, 19finlocfin 21545 . . . 4 ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
62613expib 1116 . . 3 (𝐽 ∈ Top → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
6360, 62syl 17 . 2 (𝐽 ∈ Comp → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
6459, 63impbid 202 1 (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cdif 3721  cun 3722  cin 3723  wss 3724  c0 4064  𝒫 cpw 4298  {csn 4317   cuni 4575   ciun 4655  cfv 6032  Fincfn 8110  Topctop 20919  Compccmp 21411  LocFinclocfin 21529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-en 8111  df-fin 8114  df-top 20920  df-cmp 21412  df-locfin 21532
This theorem is referenced by: (None)
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