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Theorem locfintop 21545
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 2771 . . 3 𝐽 = 𝐽
2 eqid 2771 . . 3 𝐴 = 𝐴
31, 2islocfin 21541 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐴 ∧ ∀𝑠 𝐽𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp1bi 1139 1 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  c0 4063   cuni 4575  cfv 6030  Fincfn 8113  Topctop 20918  LocFinclocfin 21528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-top 20919  df-locfin 21531
This theorem is referenced by:  lfinun  21549  locfinreflem  30247  locfinref  30248
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