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Theorem locfinnei 23382
Description: A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
locfinnei.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
locfinnei ((𝐴 ∈ (LocFinβ€˜π½) ∧ 𝑃 ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
Distinct variable groups:   𝑛,𝑠,𝐴   𝑛,𝐽   𝑃,𝑛
Allowed substitution hints:   𝑃(𝑠)   𝐽(𝑠)   𝑋(𝑛,𝑠)

Proof of Theorem locfinnei
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 locfinnei.1 . . . 4 𝑋 = βˆͺ 𝐽
2 eqid 2726 . . . 4 βˆͺ 𝐴 = βˆͺ 𝐴
31, 2islocfin 23376 . . 3 (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ 𝑋 = βˆͺ 𝐴 ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
43simp3bi 1144 . 2 (𝐴 ∈ (LocFinβ€˜π½) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
5 eleq1 2815 . . . . 5 (π‘₯ = 𝑃 β†’ (π‘₯ ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
65anbi1d 629 . . . 4 (π‘₯ = 𝑃 β†’ ((π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
76rexbidv 3172 . . 3 (π‘₯ = 𝑃 β†’ (βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ βˆƒπ‘› ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
87rspccva 3605 . 2 ((βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ∧ 𝑃 ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
94, 8sylan 579 1 ((𝐴 ∈ (LocFinβ€˜π½) ∧ 𝑃 ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   ∩ cin 3942  βˆ…c0 4317  βˆͺ cuni 4902  β€˜cfv 6537  Fincfn 8941  Topctop 22750  LocFinclocfin 23363
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-top 22751  df-locfin 23366
This theorem is referenced by:  lfinpfin  23383  lfinun  23384  locfincmp  23385  locfincf  23390
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