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Theorem locfintop 23007
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 2733 . . 3 𝐽 = 𝐽
2 eqid 2733 . . 3 𝐴 = 𝐴
31, 2islocfin 23003 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐴 ∧ ∀𝑠 𝐽𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp1bi 1146 1 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  cin 3946  c0 4321   cuni 4907  cfv 6540  Fincfn 8935  Topctop 22377  LocFinclocfin 22990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-top 22378  df-locfin 22993
This theorem is referenced by:  lfinun  23011  locfinreflem  32758  locfinref  32759
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