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Theorem locfintop 22895
Description: A locally finite cover covers a topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
Assertion
Ref Expression
locfintop (𝐴 ∈ (LocFinβ€˜π½) β†’ 𝐽 ∈ Top)

Proof of Theorem locfintop
Dummy variables 𝑛 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
2 eqid 2733 . . 3 βˆͺ 𝐴 = βˆͺ 𝐴
31, 2islocfin 22891 . 2 (𝐴 ∈ (LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ βˆͺ 𝐽 = βˆͺ 𝐴 ∧ βˆ€π‘  ∈ βˆͺ π½βˆƒπ‘› ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {π‘₯ ∈ 𝐴 ∣ (π‘₯ ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
43simp1bi 1146 1 (𝐴 ∈ (LocFinβ€˜π½) β†’ 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   ∩ cin 3913  βˆ…c0 4286  βˆͺ cuni 4869  β€˜cfv 6500  Fincfn 8889  Topctop 22265  LocFinclocfin 22878
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-top 22266  df-locfin 22881
This theorem is referenced by:  lfinun  22899  locfinreflem  32485  locfinref  32486
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