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Theorem locfinbas 20530
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 20525 . 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 1023 1  |-  ( A  e.  ( LocFin `  J
)  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740    i^i cin 3402   (/)c0 3730   U.cuni 4197   ` cfv 5581   Fincfn 7566   Topctop 19910   LocFinclocfin 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fv 5589  df-top 19914  df-locfin 20515
This theorem is referenced by:  lfinpfin  20532  lfinun  20533  locfincmp  20534  locfindis  20538  locfincf  20539
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