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Theorem locfindis 23452
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 23446 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝐢 ∈ PtFin)
2 unipw 5446 . . . . 5 βˆͺ 𝒫 𝑋 = 𝑋
32eqcomi 2734 . . . 4 𝑋 = βˆͺ 𝒫 𝑋
4 locfindis.1 . . . 4 π‘Œ = βˆͺ 𝐢
53, 4locfinbas 23444 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝑋 = π‘Œ)
61, 5jca 510 . 2 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ (𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ))
7 simpr 483 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
8 uniexg 7743 . . . . . . 7 (𝐢 ∈ PtFin β†’ βˆͺ 𝐢 ∈ V)
94, 8eqeltrid 2829 . . . . . 6 (𝐢 ∈ PtFin β†’ π‘Œ ∈ V)
109adantr 479 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ π‘Œ ∈ V)
117, 10eqeltrd 2825 . . . 4 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 ∈ V)
12 distop 22916 . . . 4 (𝑋 ∈ V β†’ 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝒫 𝑋 ∈ Top)
14 snelpwi 5439 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ {π‘₯} ∈ 𝒫 𝑋)
1514adantl 480 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
16 snidg 4658 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ π‘₯ ∈ {π‘₯})
1716adantl 480 . . . . 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 2811 . . . . . . 7 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ π‘Œ))
2019biimpa 475 . . . . . 6 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ π‘Œ)
214ptfinfin 23441 . . . . . 6 ((𝐢 ∈ PtFin ∧ π‘₯ ∈ π‘Œ) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
2218, 20, 21syl2anc 582 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
23 eleq2 2814 . . . . . . 7 (𝑦 = {π‘₯} β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ {π‘₯}))
24 ineq2 4200 . . . . . . . . . . 11 (𝑦 = {π‘₯} β†’ (𝑠 ∩ 𝑦) = (𝑠 ∩ {π‘₯}))
2524neeq1d 2990 . . . . . . . . . 10 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…))
26 disjsn 4711 . . . . . . . . . . 11 ((𝑠 ∩ {π‘₯}) = βˆ… ↔ Β¬ π‘₯ ∈ 𝑠)
2726necon2abii 2981 . . . . . . . . . 10 (π‘₯ ∈ 𝑠 ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…)
2825, 27bitr4di 288 . . . . . . . . 9 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ π‘₯ ∈ 𝑠))
2928rabbidv 3427 . . . . . . . 8 (𝑦 = {π‘₯} β†’ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} = {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠})
3029eleq1d 2810 . . . . . . 7 (𝑦 = {π‘₯} β†’ ({𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin ↔ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin))
3123, 30anbi12d 630 . . . . . 6 (𝑦 = {π‘₯} β†’ ((π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin) ↔ (π‘₯ ∈ {π‘₯} ∧ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)))
3231rspcev 3601 . . . . 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 3136 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
353, 4islocfin 23439 . . 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 1340 . 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419  Vcvv 3463   ∩ cin 3938  βˆ…c0 4318  π’« cpw 4598  {csn 4624  βˆͺ cuni 4903  β€˜cfv 6543  Fincfn 8962  Topctop 22813  PtFincptfin 23425  LocFinclocfin 23426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7869  df-1o 8485  df-en 8963  df-fin 8966  df-top 22814  df-ptfin 23428  df-locfin 23429
This theorem is referenced by: (None)
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