Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  locfinbas Structured version   Unicode version

Theorem locfinbas 26383
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  = 
U. J
locfinbas.2  |-  Y  = 
U. A
Assertion
Ref Expression
locfinbas  |-  ( A  e.  ( 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  = 
U. J
2 locfinbas.2 . . 3  |-  Y  = 
U. A
31, 2islocfin 26378 . 2  |-  ( A  e.  ( LocFin `  J
)  <->  ( J  e. 
Top  /\  X  =  Y  /\  A. s  e.  X  E. n  e.  J  ( s  e.  n  /\  { x  e.  A  |  (
x  i^i  n )  =/=  (/) }  e.  Fin ) ) )
43simp2bi 974 1  |-  ( A  e.  ( LocFin `  J
)  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711    i^i cin 3321   (/)c0 3630   U.cuni 4017   ` cfv 5456   Fincfn 7111   Topctop 16960   LocFinclocfin 26344
This theorem is referenced by:  lfinpfin  26385  locfincmp  26386  locfindis  26387  locfincf  26388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-top 16965  df-locfin 26348
  Copyright terms: Public domain W3C validator