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Theorem cmpcov 19671
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 461 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  C_  J )
2 ssexg 4593 . . . . . 6  |-  ( ( S  C_  J  /\  J  e.  Comp )  ->  S  e.  _V )
32ancoms 453 . . . . 5  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  _V )
4 elpwg 4018 . . . . 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 19670 . . . . 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 464 . . . 4  |-  ( J  e.  Comp  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
109adantr 465 . . 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 4253 . . . . . 6  |-  ( r  =  S  ->  U. r  =  U. S )
1211eqeq2d 2481 . . . . 5  |-  ( r  =  S  ->  ( X  =  U. r  <->  X  =  U. S ) )
13 pweq 4013 . . . . . . 7  |-  ( r  =  S  ->  ~P r  =  ~P S
)
1413ineq1d 3699 . . . . . 6  |-  ( r  =  S  ->  ( ~P r  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514rexeqdv 3065 . . . . 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 320 . . . 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 3210 . . 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 1193 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   Fincfn 7516   Topctop 19177   Compccmp 19668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-cmp 19669
This theorem is referenced by:  cmpcov2  19672  cncmp  19674  discmp  19680  cmpcld  19684  sscmp  19687  alexsubALTlem1  20298  ptcmplem3  20305  lebnum  21215  comppfsc  29795  heibor1  29925
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