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Theorem locfinbas 21828
Description: A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
locfinbas.1 𝑋 = 𝐽
locfinbas.2 𝑌 = 𝐴
Assertion
Ref Expression
locfinbas (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

Proof of Theorem locfinbas
Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfinbas.1 . . 3 𝑋 = 𝐽
2 locfinbas.2 . . 3 𝑌 = 𝐴
31, 2islocfin 21823 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠𝑋𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp2bi 1126 1 (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  wne 2964  wral 3085  wrex 3086  {crab 3089  cin 3827  c0 4177   cuni 4710  cfv 6186  Fincfn 8302  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 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277
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 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fv 6194  df-top 21200  df-locfin 21813
This theorem is referenced by:  lfinpfin  21830  lfinun  21831  locfincmp  21832  locfindis  21836  locfincf  21837
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