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Theorem locfindis 22897
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 22891 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝐢 ∈ PtFin)
2 unipw 5412 . . . . 5 βˆͺ 𝒫 𝑋 = 𝑋
32eqcomi 2746 . . . 4 𝑋 = βˆͺ 𝒫 𝑋
4 locfindis.1 . . . 4 π‘Œ = βˆͺ 𝐢
53, 4locfinbas 22889 . . 3 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ 𝑋 = π‘Œ)
61, 5jca 513 . 2 (𝐢 ∈ (LocFinβ€˜π’« 𝑋) β†’ (𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ))
7 simpr 486 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 = π‘Œ)
8 uniexg 7682 . . . . . . 7 (𝐢 ∈ PtFin β†’ βˆͺ 𝐢 ∈ V)
94, 8eqeltrid 2842 . . . . . 6 (𝐢 ∈ PtFin β†’ π‘Œ ∈ V)
109adantr 482 . . . . 5 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ π‘Œ ∈ V)
117, 10eqeltrd 2838 . . . 4 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝑋 ∈ V)
12 distop 22361 . . . 4 (𝑋 ∈ V β†’ 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ 𝒫 𝑋 ∈ Top)
14 snelpwi 5405 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ {π‘₯} ∈ 𝒫 𝑋)
1514adantl 483 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
16 snidg 4625 . . . . . 6 (π‘₯ ∈ 𝑋 β†’ π‘₯ ∈ {π‘₯})
1716adantl 483 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ {π‘₯})
18 simpll 766 . . . . . 6 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ PtFin)
197eleq2d 2824 . . . . . . 7 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ π‘Œ))
2019biimpa 478 . . . . . 6 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ π‘Œ)
214ptfinfin 22886 . . . . . 6 ((𝐢 ∈ PtFin ∧ π‘₯ ∈ π‘Œ) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
2218, 20, 21syl2anc 585 . . . . 5 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)
23 eleq2 2827 . . . . . . 7 (𝑦 = {π‘₯} β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ {π‘₯}))
24 ineq2 4171 . . . . . . . . . . 11 (𝑦 = {π‘₯} β†’ (𝑠 ∩ 𝑦) = (𝑠 ∩ {π‘₯}))
2524neeq1d 3004 . . . . . . . . . 10 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…))
26 disjsn 4677 . . . . . . . . . . 11 ((𝑠 ∩ {π‘₯}) = βˆ… ↔ Β¬ π‘₯ ∈ 𝑠)
2726necon2abii 2995 . . . . . . . . . 10 (π‘₯ ∈ 𝑠 ↔ (𝑠 ∩ {π‘₯}) β‰  βˆ…)
2825, 27bitr4di 289 . . . . . . . . 9 (𝑦 = {π‘₯} β†’ ((𝑠 ∩ 𝑦) β‰  βˆ… ↔ π‘₯ ∈ 𝑠))
2928rabbidv 3418 . . . . . . . 8 (𝑦 = {π‘₯} β†’ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} = {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠})
3029eleq1d 2823 . . . . . . 7 (𝑦 = {π‘₯} β†’ ({𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin ↔ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin))
3123, 30anbi12d 632 . . . . . 6 (𝑦 = {π‘₯} β†’ ((π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin) ↔ (π‘₯ ∈ {π‘₯} ∧ {𝑠 ∈ 𝐢 ∣ π‘₯ ∈ 𝑠} ∈ Fin)))
3231rspcev 3584 . . . . 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 836 . . . 4 (((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
3433ralrimiva 3144 . . 3 ((𝐢 ∈ PtFin ∧ 𝑋 = π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝒫 𝑋(π‘₯ ∈ 𝑦 ∧ {𝑠 ∈ 𝐢 ∣ (𝑠 ∩ 𝑦) β‰  βˆ…} ∈ Fin))
353, 4islocfin 22884 . . 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 1344 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   ∩ cin 3914  βˆ…c0 4287  π’« cpw 4565  {csn 4591  βˆͺ cuni 4870  β€˜cfv 6501  Fincfn 8890  Topctop 22258  PtFincptfin 22870  LocFinclocfin 22871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894  df-top 22259  df-ptfin 22873  df-locfin 22874
This theorem is referenced by: (None)
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