MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  locfinbas Structured version   Visualization version   GIF version

Theorem locfinbas 23530
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 23525 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠𝑋𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp2bi 1147 1 (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cin 3950  c0 4333   cuni 4907  cfv 6561  Fincfn 8985  Topctop 22899  LocFinclocfin 23512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-locfin 23515
This theorem is referenced by:  lfinpfin  23532  lfinun  23533  locfincmp  23534  locfindis  23538  locfincf  23539
  Copyright terms: Public domain W3C validator