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Theorem locfindis 22235
 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 22229 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2 unipw 5314 . . . . 5 𝒫 𝑋 = 𝑋
32eqcomi 2767 . . . 4 𝑋 = 𝒫 𝑋
4 locfindis.1 . . . 4 𝑌 = 𝐶
53, 4locfinbas 22227 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
61, 5jca 515 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7 simpr 488 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8 uniexg 7469 . . . . . . 7 (𝐶 ∈ PtFin → 𝐶 ∈ V)
94, 8eqeltrid 2856 . . . . . 6 (𝐶 ∈ PtFin → 𝑌 ∈ V)
109adantr 484 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
117, 10eqeltrd 2852 . . . 4 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12 distop 21700 . . . 4 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14 snelpwi 5308 . . . . . 6 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
1514adantl 485 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
16 snidg 4559 . . . . . 6 (𝑥𝑋𝑥 ∈ {𝑥})
1716adantl 485 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
18 simpll 766 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐶 ∈ PtFin)
197eleq2d 2837 . . . . . . 7 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥𝑋𝑥𝑌))
2019biimpa 480 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑌)
214ptfinfin 22224 . . . . . 6 ((𝐶 ∈ PtFin ∧ 𝑥𝑌) → {𝑠𝐶𝑥𝑠} ∈ Fin)
2218, 20, 21syl2anc 587 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐶𝑥𝑠} ∈ Fin)
23 eleq2 2840 . . . . . . 7 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
24 ineq2 4113 . . . . . . . . . . 11 (𝑦 = {𝑥} → (𝑠𝑦) = (𝑠 ∩ {𝑥}))
2524neeq1d 3010 . . . . . . . . . 10 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26 disjsn 4607 . . . . . . . . . . 11 ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝑠)
2726necon2abii 3001 . . . . . . . . . 10 (𝑥𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
2825, 27bitr4di 292 . . . . . . . . 9 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ 𝑥𝑠))
2928rabbidv 3392 . . . . . . . 8 (𝑦 = {𝑥} → {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} = {𝑠𝐶𝑥𝑠})
3029eleq1d 2836 . . . . . . 7 (𝑦 = {𝑥} → ({𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin ↔ {𝑠𝐶𝑥𝑠} ∈ Fin))
3123, 30anbi12d 633 . . . . . 6 (𝑦 = {𝑥} → ((𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)))
3231rspcev 3543 . . . . 5 (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3315, 17, 22, 32syl12anc 835 . . . 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 22222 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1340 . 2 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
376, 36impbii 212 1 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∩ cin 3859  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801  ‘cfv 6339  Fincfn 8532  Topctop 21598  PtFincptfin 22208  LocFinclocfin 22209 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7585  df-1o 8117  df-en 8533  df-fin 8536  df-top 21599  df-ptfin 22211  df-locfin 22212 This theorem is referenced by: (None)
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