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Theorem cmpcov 18951
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 458 . . . 4  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  C_  J )
2 ssexg 4435 . . . . . 6  |-  ( ( S  C_  J  /\  J  e.  Comp )  ->  S  e.  _V )
32ancoms 450 . . . . 5  |-  ( ( J  e.  Comp  /\  S  C_  J )  ->  S  e.  _V )
4 elpwg 3865 . . . . 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 18950 . . . . 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 461 . . . 4  |-  ( J  e.  Comp  ->  A. r  e.  ~P  J ( X  =  U. r  ->  E. s  e.  ( ~P r  i^i  Fin ) X  =  U. s
) )
109adantr 462 . . 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 4096 . . . . . 6  |-  ( r  =  S  ->  U. r  =  U. S )
1211eqeq2d 2452 . . . . 5  |-  ( r  =  S  ->  ( X  =  U. r  <->  X  =  U. S ) )
13 pweq 3860 . . . . . . 7  |-  ( r  =  S  ->  ~P r  =  ~P S
)
1413ineq1d 3548 . . . . . 6  |-  ( r  =  S  ->  ( ~P r  i^i  Fin )  =  ( ~P S  i^i  Fin ) )
1514rexeqdv 2922 . . . . 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 3066 . . 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 1179 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   Fincfn 7306   Topctop 18457   Compccmp 18948
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 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-in 3332  df-ss 3339  df-pw 3859  df-uni 4089  df-cmp 18949
This theorem is referenced by:  cmpcov2  18952  cncmp  18954  discmp  18960  cmpcld  18964  sscmp  18967  alexsubALTlem1  19578  ptcmplem3  19585  lebnum  20495  comppfsc  28504  heibor1  28634
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