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Theorem locfindis 23033
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 π‘Œ = βˆͺ 𝐢
Assertion
Ref Expression
locfindis (𝐢 ∈ (LocFinβ€˜π’« 𝑋) ↔ (𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ))

Proof of Theorem locfindis
Dummy variables π‘₯ 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfinpfin 23027 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝐢 ∈ PtFin)
2 unipw 5450 . . . . 5 βˆͺ 𝒫 𝑋 = 𝑋
32eqcomi 2741 . . . 4 𝑋 = βˆͺ 𝒫 𝑋
4 locfindis.1 . . . 4 π‘Œ = βˆͺ 𝐢
53, 4locfinbas 23025 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝑋 = π‘Œ)
61, 5jca 512 . 2 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ (𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ))
7 simpr 485 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
8 uniexg 7729 . . . . . . 7 (𝐢 ∈ PtFin β†’ βˆͺ 𝐢 ∈ V)
94, 8eqeltrid 2837 . . . . . 6 (𝐢 ∈ PtFin β†’ π‘Œ ∈ V)
109adantr 481 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ π‘Œ ∈ V)
117, 10eqeltrd 2833 . . . 4 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 ∈ V)
12 distop 22497 . . . 4 (𝑋 ∈ V β†’ 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝒫 𝑋 ∈ Top)
14 snelpwi 5443 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ {π‘₯} ∈ 𝒫 𝑋)
1514adantl 482 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
16 snidg 4662 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ π‘₯ ∈ {π‘₯})
1716adantl 482 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ {π‘₯})
18 simpll 765 . . . . . 6 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ PtFin)
197eleq2d 2819 . . . . . . 7 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ π‘Œ))
2019biimpa 477 . . . . . 6 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ π‘Œ)
214ptfinfin 23022 . . . . . 6 ((𝐢 ∈ PtFin ∧ π‘₯ ∈ π‘Œ) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
2218, 20, 21syl2anc 584 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
23 eleq2 2822 . . . . . . 7 (𝑦 = {π‘₯} β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ {π‘₯}))
24 ineq2 4206 . . . . . . . . . . 11 (𝑦 = {π‘₯} β†’ (𝑠 ∩ 𝑦) = (𝑠 ∩ {π‘₯}))
2524neeq1d 3000 . . . . . . . . . 10 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…))
26 disjsn 4715 . . . . . . . . . . 11 ((𝑠 ∩ {π‘₯}) = βˆ… ↔ Β¬ π‘₯ ∈ 𝑠)
2726necon2abii 2991 . . . . . . . . . 10 (π‘₯ ∈ 𝑠 ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…)
2825, 27bitr4di 288 . . . . . . . . 9 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ π‘₯ ∈ 𝑠))
2928rabbidv 3440 . . . . . . . 8 (𝑦 = {π‘₯} β†’ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} = {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠})
3029eleq1d 2818 . . . . . . 7 (𝑦 = {π‘₯} β†’ ({𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin ↔ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin))
3123, 30anbi12d 631 . . . . . 6 (𝑦 = {π‘₯} β†’ ((π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin) ↔ (π‘₯ ∈ {π‘₯} ∧ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)))
3231rspcev 3612 . . . . 5 (({π‘₯} ∈ 𝒫 𝑋 ∧ (π‘₯ ∈ {π‘₯} ∧ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)) β†’ βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
3315, 17, 22, 32syl12anc 835 . . . 4 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
3433ralrimiva 3146 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
353, 4islocfin 23020 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1343 . 2 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝐢 ∈ (LocFinβ€˜π’« 𝑋))
376, 36impbii 208 1 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) ↔ (𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  Fincfn 8938  Topctop 22394  PtFincptfin 23006  LocFinclocfin 23007
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-en 8939  df-fin 8942  df-top 22395  df-ptfin 23009  df-locfin 23010
This theorem is referenced by: (None)
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