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

Theorem locfintop 21534
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 2806 . . 3 𝐽 = 𝐽
2 eqid 2806 . . 3 𝐴 = 𝐴
31, 2islocfin 21530 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝐽 = 𝐴 ∧ ∀𝑠 𝐽𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp1bi 1168 1 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wne 2978  wral 3096  wrex 3097  {crab 3100  cin 3768  c0 4116   cuni 4630  cfv 6097  Fincfn 8188  Topctop 20907  LocFinclocfin 21517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fv 6105  df-top 20908  df-locfin 21520
This theorem is referenced by:  lfinun  21538  locfinreflem  30228  locfinref  30229
  Copyright terms: Public domain W3C validator