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Theorem cmpcov 17132
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 447 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  C_  J )
2 ssexg 4176 . . . . . 6  |-  ( ( S  C_  J  /\  J  e.  Comp )  ->  S  e.  _V )
32ancoms 439 . . . . 5  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  _V )
4 elpwg 3645 . . . . 5  |-  ( S  e.  _V  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
53, 4syl 15 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( S  e.  ~P J  <->  S 
C_  J ) )
61, 5mpbird 223 . . 3  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  ~P J )
7 iscmp.1 . . . . . 6  |-  X  = 
U. J
87iscmp 17131 . . . . 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 450 . . . 4  |-  ( J  e.  Comp  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
109adantr 451 . . 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 3852 . . . . . 6  |-  ( r  =  S  ->  U. r  =  U. S )
1211eqeq2d 2307 . . . . 5  |-  ( r  =  S  ->  ( X  =  U. r  <->  X  =  U. S ) )
13 pweq 3641 . . . . . . 7  |-  ( r  =  S  ->  ~P r  =  ~P S
)
1413ineq1d 3382 . . . . . 6  |-  ( r  =  S  ->  ( ~P r  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514rexeqdv 2756 . . . . 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 311 . . . 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 2893 . . 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 56 . 2  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  ( X  =  U. S  ->  E. s  e.  ( ~P S  i^i  Fin ) X  =  U. s
) )
19183impia 1148 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   Fincfn 6879   Topctop 16647   Compccmp 17129
This theorem is referenced by:  cmpcov2  17133  cncmp  17135  discmp  17141  cmpcld  17145  sscmp  17148  alexsubALTlem1  17757  ptcmplem3  17764  lebnum  18478  comppfsc  26410  heibor1  26637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-cmp 17130
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