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Theorem locfintop 22057
Description: A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
Assertion
Ref Expression
locfintop (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)

Proof of Theorem locfintop
Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . 3 𝐽 = 𝐽
2 eqid 2818 . . 3 𝐴 = 𝐴
31, 2islocfin 22053 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐴 ∧ ∀𝑠 𝐽𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp1bi 1137 1 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  c0 4288   cuni 4830  cfv 6348  Fincfn 8497  Topctop 21429  LocFinclocfin 22040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-locfin 22043
This theorem is referenced by:  lfinun  22061  locfinreflem  31003  locfinref  31004
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