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Theorem locfinnei 23026
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 2732 . . . 4 βˆͺ 𝐴 = βˆͺ 𝐴
31, 2islocfin 23020 . . 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 1147 . 2 (𝐴 ∈ (LocFinβ€˜π½) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘› ∈ 𝐽 (π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
5 eleq1 2821 . . . . 5 (π‘₯ = 𝑃 β†’ (π‘₯ ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
65anbi1d 630 . . . 4 (π‘₯ = 𝑃 β†’ ((π‘₯ ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) ↔ (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
76rexbidv 3178 . . 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 3611 . 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 580 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   ∩ cin 3947  βˆ…c0 4322  βˆͺ cuni 4908  β€˜cfv 6543  Fincfn 8938  Topctop 22394  LocFinclocfin 23007
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22395  df-locfin 23010
This theorem is referenced by:  lfinpfin  23027  lfinun  23028  locfincmp  23029  locfincf  23034
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