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Theorem cmpcov 20056
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 459 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  C_  J )
2 ssexg 4583 . . . . . 6  |-  ( ( S  C_  J  /\  J  e.  Comp )  ->  S  e.  _V )
32ancoms 451 . . . . 5  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  _V )
4 elpwg 4007 . . . . 5  |-  ( S  e.  _V  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
53, 4syl 16 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
61, 5mpbird 232 . . 3  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  ~P J )
7 iscmp.1 . . . . . 6  |-  X  = 
U. J
87iscmp 20055 . . . . 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 462 . . . 4  |-  ( J  e.  Comp  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
109adantr 463 . . 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 4243 . . . . . 6  |-  ( r  =  S  ->  U. r  =  U. S )
1211eqeq2d 2468 . . . . 5  |-  ( r  =  S  ->  ( X  =  U. r  <->  X  =  U. S ) )
13 pweq 4002 . . . . . . 7  |-  ( r  =  S  ->  ~P r  =  ~P S
)
1413ineq1d 3685 . . . . . 6  |-  ( r  =  S  ->  ( ~P r  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514rexeqdv 3058 . . . . 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 318 . . . 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 3203 . . 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 60 . 2  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( X  =  U. S  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
) )
19183impia 1191 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   Fincfn 7509   Topctop 19561   Compccmp 20053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001  df-uni 4236  df-cmp 20054
This theorem is referenced by:  cmpcov2  20057  cncmp  20059  discmp  20065  cmpcld  20069  sscmp  20072  comppfsc  20199  alexsubALTlem1  20713  ptcmplem3  20720  lebnum  21630  heibor1  30546
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