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Theorem locfindis 23539
Description: The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
locfindis.1 𝑌 = 𝐶
Assertion
Ref Expression
locfindis (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))

Proof of Theorem locfindis
Dummy variables 𝑥 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfinpfin 23533 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2 unipw 5454 . . . . 5 𝒫 𝑋 = 𝑋
32eqcomi 2745 . . . 4 𝑋 = 𝒫 𝑋
4 locfindis.1 . . . 4 𝑌 = 𝐶
53, 4locfinbas 23531 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
61, 5jca 511 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7 simpr 484 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8 uniexg 7761 . . . . . . 7 (𝐶 ∈ PtFin → 𝐶 ∈ V)
94, 8eqeltrid 2844 . . . . . 6 (𝐶 ∈ PtFin → 𝑌 ∈ V)
109adantr 480 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
117, 10eqeltrd 2840 . . . 4 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12 distop 23003 . . . 4 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14 snelpwi 5447 . . . . . 6 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
1514adantl 481 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
16 snidg 4659 . . . . . 6 (𝑥𝑋𝑥 ∈ {𝑥})
1716adantl 481 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
18 simpll 766 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐶 ∈ PtFin)
197eleq2d 2826 . . . . . . 7 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥𝑋𝑥𝑌))
2019biimpa 476 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑌)
214ptfinfin 23528 . . . . . 6 ((𝐶 ∈ PtFin ∧ 𝑥𝑌) → {𝑠𝐶𝑥𝑠} ∈ Fin)
2218, 20, 21syl2anc 584 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐶𝑥𝑠} ∈ Fin)
23 eleq2 2829 . . . . . . 7 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
24 ineq2 4213 . . . . . . . . . . 11 (𝑦 = {𝑥} → (𝑠𝑦) = (𝑠 ∩ {𝑥}))
2524neeq1d 2999 . . . . . . . . . 10 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26 disjsn 4710 . . . . . . . . . . 11 ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝑠)
2726necon2abii 2990 . . . . . . . . . 10 (𝑥𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
2825, 27bitr4di 289 . . . . . . . . 9 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ 𝑥𝑠))
2928rabbidv 3443 . . . . . . . 8 (𝑦 = {𝑥} → {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} = {𝑠𝐶𝑥𝑠})
3029eleq1d 2825 . . . . . . 7 (𝑦 = {𝑥} → ({𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin ↔ {𝑠𝐶𝑥𝑠} ∈ Fin))
3123, 30anbi12d 632 . . . . . 6 (𝑦 = {𝑥} → ((𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)))
3231rspcev 3621 . . . . 5 (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3315, 17, 22, 32syl12anc 836 . . . 4 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3433ralrimiva 3145 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
353, 4islocfin 23526 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1343 . 2 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
376, 36impbii 209 1 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  cin 3949  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906  cfv 6560  Fincfn 8986  Topctop 22900  PtFincptfin 23512  LocFinclocfin 23513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-en 8987  df-fin 8990  df-top 22901  df-ptfin 23515  df-locfin 23516
This theorem is referenced by: (None)
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