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Theorem locfindis 22073
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 22067 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝐶 ∈ PtFin)
2 unipw 5339 . . . . 5 𝒫 𝑋 = 𝑋
32eqcomi 2835 . . . 4 𝑋 = 𝒫 𝑋
4 locfindis.1 . . . 4 𝑌 = 𝐶
53, 4locfinbas 22065 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) → 𝑋 = 𝑌)
61, 5jca 512 . 2 (𝐶 ∈ (LocFin‘𝒫 𝑋) → (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
7 simpr 485 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
8 uniexg 7461 . . . . . . 7 (𝐶 ∈ PtFin → 𝐶 ∈ V)
94, 8eqeltrid 2922 . . . . . 6 (𝐶 ∈ PtFin → 𝑌 ∈ V)
109adantr 481 . . . . 5 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑌 ∈ V)
117, 10eqeltrd 2918 . . . 4 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝑋 ∈ V)
12 distop 21538 . . . 4 (𝑋 ∈ V → 𝒫 𝑋 ∈ Top)
1311, 12syl 17 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝒫 𝑋 ∈ Top)
14 snelpwi 5333 . . . . . 6 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
1514adantl 482 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
16 snidg 4596 . . . . . 6 (𝑥𝑋𝑥 ∈ {𝑥})
1716adantl 482 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
18 simpll 763 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐶 ∈ PtFin)
197eleq2d 2903 . . . . . . 7 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → (𝑥𝑋𝑥𝑌))
2019biimpa 477 . . . . . 6 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑌)
214ptfinfin 22062 . . . . . 6 ((𝐶 ∈ PtFin ∧ 𝑥𝑌) → {𝑠𝐶𝑥𝑠} ∈ Fin)
2218, 20, 21syl2anc 584 . . . . 5 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐶𝑥𝑠} ∈ Fin)
23 eleq2 2906 . . . . . . 7 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
24 ineq2 4187 . . . . . . . . . . 11 (𝑦 = {𝑥} → (𝑠𝑦) = (𝑠 ∩ {𝑥}))
2524neeq1d 3080 . . . . . . . . . 10 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ (𝑠 ∩ {𝑥}) ≠ ∅))
26 disjsn 4646 . . . . . . . . . . 11 ((𝑠 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝑠)
2726necon2abii 3071 . . . . . . . . . 10 (𝑥𝑠 ↔ (𝑠 ∩ {𝑥}) ≠ ∅)
2825, 27syl6bbr 290 . . . . . . . . 9 (𝑦 = {𝑥} → ((𝑠𝑦) ≠ ∅ ↔ 𝑥𝑠))
2928rabbidv 3486 . . . . . . . 8 (𝑦 = {𝑥} → {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} = {𝑠𝐶𝑥𝑠})
3029eleq1d 2902 . . . . . . 7 (𝑦 = {𝑥} → ({𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin ↔ {𝑠𝐶𝑥𝑠} ∈ Fin))
3123, 30anbi12d 630 . . . . . 6 (𝑦 = {𝑥} → ((𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)))
3231rspcev 3627 . . . . 5 (({𝑥} ∈ 𝒫 𝑋 ∧ (𝑥 ∈ {𝑥} ∧ {𝑠𝐶𝑥𝑠} ∈ Fin)) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3315, 17, 22, 32syl12anc 834 . . . 4 (((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
3433ralrimiva 3187 . . 3 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin))
353, 4islocfin 22060 . . 3 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝒫 𝑋 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑦 ∈ 𝒫 𝑋(𝑥𝑦 ∧ {𝑠𝐶 ∣ (𝑠𝑦) ≠ ∅} ∈ Fin)))
3613, 7, 34, 35syl3anbrc 1337 . 2 ((𝐶 ∈ PtFin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝒫 𝑋))
376, 36impbii 210 1 (𝐶 ∈ (LocFin‘𝒫 𝑋) ↔ (𝐶 ∈ PtFin ∧ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  cin 3939  c0 4295  𝒫 cpw 4542  {csn 4564   cuni 4837  cfv 6354  Fincfn 8503  Topctop 21436  PtFincptfin 22046  LocFinclocfin 22047
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7574  df-er 8284  df-en 8504  df-fin 8507  df-top 21437  df-ptfin 22049  df-locfin 22050
This theorem is referenced by: (None)
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