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Theorem locfinbas 29760
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 29755 . 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 1007 1  |-  ( A  e.  ( LocFin `  J
)  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3468   (/)c0 3778   U.cuni 4238   ` cfv 5579   Fincfn 7506   Topctop 19154   LocFinclocfin 29721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-top 19159  df-locfin 29725
This theorem is referenced by:  lfinpfin  29762  locfincmp  29763  locfindis  29764  locfincf  29765
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