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Theorem locfinnei 21829
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 = 𝐽
Assertion
Ref Expression
locfinnei ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃𝑋) → ∃𝑛𝐽 (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ 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 𝑥 is distinct from all other variables.
StepHypRef Expression
1 locfinnei.1 . . . 4 𝑋 = 𝐽
2 eqid 2772 . . . 4 𝐴 = 𝐴
31, 2islocfin 21823 . . 3 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐴 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
43simp3bi 1127 . 2 (𝐴 ∈ (LocFin‘𝐽) → ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
5 eleq1 2847 . . . . 5 (𝑥 = 𝑃 → (𝑥𝑛𝑃𝑛))
65anbi1d 620 . . . 4 (𝑥 = 𝑃 → ((𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
76rexbidv 3236 . . 3 (𝑥 = 𝑃 → (∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
87rspccva 3528 . 2 ((∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ∧ 𝑃𝑋) → ∃𝑛𝐽 (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
94, 8sylan 572 1 ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃𝑋) → ∃𝑛𝐽 (𝑃𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wne 2961  wral 3082  wrex 3083  {crab 3086  cin 3822  c0 4172   cuni 4706  cfv 6182  Fincfn 8300  Topctop 21199  LocFinclocfin 21810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fv 6190  df-top 21200  df-locfin 21813
This theorem is referenced by:  lfinpfin  21830  lfinun  21831  locfincmp  21832  locfincf  21837
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