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Theorem cmpcov 20404
Description: An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)
Hypothesis
Ref Expression
iscmp.1  |-  X  = 
U. J
Assertion
Ref Expression
cmpcov  |-  ( ( J  e.  Comp  /\  S  C_  J  /\  X  = 
U. S )  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
)
Distinct variable groups:    J, s    S, s
Allowed substitution hint:    X( s)

Proof of Theorem cmpcov
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 simpr 463 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  C_  J )
2 ssexg 4549 . . . . . 6  |-  ( ( S  C_  J  /\  J  e.  Comp )  ->  S  e.  _V )
32ancoms 455 . . . . 5  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  _V )
4 elpwg 3959 . . . . 5  |-  ( S  e.  _V  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
53, 4syl 17 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
61, 5mpbird 236 . . 3  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  ~P J )
7 iscmp.1 . . . . . 6  |-  X  = 
U. J
87iscmp 20403 . . . . 5  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) ) )
98simprbi 466 . . . 4  |-  ( J  e.  Comp  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
109adantr 467 . . 3  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
11 unieq 4206 . . . . . 6  |-  ( r  =  S  ->  U. r  =  U. S )
1211eqeq2d 2461 . . . . 5  |-  ( r  =  S  ->  ( X  =  U. r  <->  X  =  U. S ) )
13 pweq 3954 . . . . . . 7  |-  ( r  =  S  ->  ~P r  =  ~P S
)
1413ineq1d 3633 . . . . . 6  |-  ( r  =  S  ->  ( ~P r  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514rexeqdv 2994 . . . . 5  |-  ( r  =  S  ->  ( E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s  <->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
) )
1612, 15imbi12d 322 . . . 4  |-  ( r  =  S  ->  (
( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s )  <->  ( X  =  U. S  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s ) ) )
1716rspcv 3146 . . 3  |-  ( S  e.  ~P J  -> 
( A. r  e. 
~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
)  ->  ( X  =  U. S  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s ) ) )
186, 10, 17sylc 62 . 2  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( X  =  U. S  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
) )
19183impia 1205 1  |-  ( ( J  e.  Comp  /\  S  C_  J  /\  X  = 
U. S )  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   Fincfn 7569   Topctop 19917   Compccmp 20401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953  df-uni 4199  df-cmp 20402
This theorem is referenced by:  cmpcov2  20405  cncmp  20407  discmp  20413  cmpcld  20417  sscmp  20420  comppfsc  20547  alexsubALTlem1  21062  ptcmplem3  21069  lebnum  21995  heibor1  32142
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