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Theorem locfindis 21613
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 21607 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2 unipw 5074 . . . . 5 𝒫 𝑋 = 𝑋
32eqcomi 2774 . . . 4 𝑋 = 𝒫 𝑋
4 locfindis.1 . . . 4 𝑌 = 𝐶
53, 4locfinbas 21605 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
61, 5jca 507 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7 simpr 477 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8 uniexg 7153 . . . . . . 7 (𝐶 ∈ PtFin → 𝐶 ∈ V)
94, 8syl5eqel 2848 . . . . . 6 (𝐶 ∈ PtFin → 𝑌 ∈ V)
109adantr 472 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
117, 10eqeltrd 2844 . . . 4 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12 distop 21079 . . . 4 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14 snelpwi 5068 . . . . . 6 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
1514adantl 473 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
16 snidg 4364 . . . . . 6 (𝑥𝑋𝑥 ∈ {𝑥})
1716adantl 473 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
18 simpll 783 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐶 ∈ PtFin)
197eleq2d 2830 . . . . . . 7 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥𝑋𝑥𝑌))
2019biimpa 468 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑌)
214ptfinfin 21602 . . . . . 6 ((𝐶 ∈ PtFin ∧ 𝑥𝑌) → {𝑠𝐶𝑥𝑠} ∈ Fin)
2218, 20, 21syl2anc 579 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐶𝑥𝑠} ∈ Fin)
23 eleq2 2833 . . . . . . 7 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
24 ineq2 3970 . . . . . . . . . . 11 (𝑦 = {𝑥} → (𝑠𝑦) = (𝑠 ∩ {𝑥}))
2524neeq1d 2996 . . . . . . . . . 10 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26 disjsn 4402 . . . . . . . . . . 11 ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝑠)
2726necon2abii 2987 . . . . . . . . . 10 (𝑥𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
2825, 27syl6bbr 280 . . . . . . . . 9 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ 𝑥𝑠))
2928rabbidv 3338 . . . . . . . 8 (𝑦 = {𝑥} → {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} = {𝑠𝐶𝑥𝑠})
3029eleq1d 2829 . . . . . . 7 (𝑦 = {𝑥} → ({𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin ↔ {𝑠𝐶𝑥𝑠} ∈ Fin))
3123, 30anbi12d 624 . . . . . 6 (𝑦 = {𝑥} → ((𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)))
3231rspcev 3461 . . . . 5 (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3315, 17, 22, 32syl12anc 865 . . . 4 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3433ralrimiva 3113 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
353, 4islocfin 21600 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1443 . 2 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
376, 36impbii 200 1 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cin 3731  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594  cfv 6068  Fincfn 8160  Topctop 20977  PtFincptfin 21586  LocFinclocfin 21587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-er 7947  df-en 8161  df-fin 8164  df-top 20978  df-ptfin 21589  df-locfin 21590
This theorem is referenced by: (None)
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