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Theorem locfindis 21554
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 21548 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2 unipw 5047 . . . . 5 𝒫 𝑋 = 𝑋
32eqcomi 2780 . . . 4 𝑋 = 𝒫 𝑋
4 locfindis.1 . . . 4 𝑌 = 𝐶
53, 4locfinbas 21546 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
61, 5jca 501 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7 simpr 471 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8 uniexg 7106 . . . . . . 7 (𝐶 ∈ PtFin → 𝐶 ∈ V)
94, 8syl5eqel 2854 . . . . . 6 (𝐶 ∈ PtFin → 𝑌 ∈ V)
109adantr 466 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
117, 10eqeltrd 2850 . . . 4 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12 distop 21020 . . . 4 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14 snelpwi 5041 . . . . . 6 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
1514adantl 467 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
16 snidg 4346 . . . . . 6 (𝑥𝑋𝑥 ∈ {𝑥})
1716adantl 467 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
18 simpll 750 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐶 ∈ PtFin)
197eleq2d 2836 . . . . . . 7 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥𝑋𝑥𝑌))
2019biimpa 462 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑌)
214ptfinfin 21543 . . . . . 6 ((𝐶 ∈ PtFin ∧ 𝑥𝑌) → {𝑠𝐶𝑥𝑠} ∈ Fin)
2218, 20, 21syl2anc 573 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐶𝑥𝑠} ∈ Fin)
23 eleq2 2839 . . . . . . 7 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
24 ineq2 3959 . . . . . . . . . . 11 (𝑦 = {𝑥} → (𝑠𝑦) = (𝑠 ∩ {𝑥}))
2524neeq1d 3002 . . . . . . . . . 10 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26 disjsn 4384 . . . . . . . . . . 11 ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝑠)
2726necon2abii 2993 . . . . . . . . . 10 (𝑥𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
2825, 27syl6bbr 278 . . . . . . . . 9 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ 𝑥𝑠))
2928rabbidv 3339 . . . . . . . 8 (𝑦 = {𝑥} → {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} = {𝑠𝐶𝑥𝑠})
3029eleq1d 2835 . . . . . . 7 (𝑦 = {𝑥} → ({𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin ↔ {𝑠𝐶𝑥𝑠} ∈ Fin))
3123, 30anbi12d 616 . . . . . 6 (𝑦 = {𝑥} → ((𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)))
3231rspcev 3460 . . . . 5 (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3315, 17, 22, 32syl12anc 1474 . . . 4 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3433ralrimiva 3115 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
353, 4islocfin 21541 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1428 . 2 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
376, 36impbii 199 1 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cin 3722  c0 4063  𝒫 cpw 4298  {csn 4317   cuni 4575  cfv 6030  Fincfn 8113  Topctop 20918  PtFincptfin 21527  LocFinclocfin 21528
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 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  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-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7217  df-er 7900  df-en 8114  df-fin 8117  df-top 20919  df-ptfin 21530  df-locfin 21531
This theorem is referenced by: (None)
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