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Theorem iscmp 17405
Description: The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
iscmp.1  |-  X  = 
U. J
Assertion
Ref Expression
iscmp  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. y  e.  ~P  J ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i  Fin ) X  =  U. z
) ) )
Distinct variable group:    y, z, J
Allowed substitution hints:    X( y, z)

Proof of Theorem iscmp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3762 . . 3  |-  ( x  =  J  ->  ~P x  =  ~P J
)
2 unieq 3984 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
3 iscmp.1 . . . . . 6  |-  X  = 
U. J
42, 3syl6eqr 2454 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
54eqeq1d 2412 . . . 4  |-  ( x  =  J  ->  ( U. x  =  U. y 
<->  X  =  U. y
) )
64eqeq1d 2412 . . . . 5  |-  ( x  =  J  ->  ( U. x  =  U. z 
<->  X  =  U. z
) )
76rexbidv 2687 . . . 4  |-  ( x  =  J  ->  ( E. z  e.  ( ~P y  i^i  Fin ) U. x  =  U. z 
<->  E. z  e.  ( ~P y  i^i  Fin ) X  =  U. z ) )
85, 7imbi12d 312 . . 3  |-  ( x  =  J  ->  (
( U. x  = 
U. y  ->  E. z  e.  ( ~P y  i^i 
Fin ) U. x  =  U. z )  <->  ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i 
Fin ) X  = 
U. z ) ) )
91, 8raleqbidv 2876 . 2  |-  ( x  =  J  ->  ( A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ( ~P y  i^i 
Fin ) U. x  =  U. z )  <->  A. y  e.  ~P  J ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i  Fin ) X  =  U. z
) ) )
10 df-cmp 17404 . 2  |-  Comp  =  { x  e.  Top  | 
A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ( ~P y  i^i 
Fin ) U. x  =  U. z ) }
119, 10elrab2 3054 1  |-  ( J  e.  Comp  <->  ( J  e. 
Top  /\  A. y  e.  ~P  J ( X  =  U. y  ->  E. z  e.  ( ~P y  i^i  Fin ) X  =  U. z
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    i^i cin 3279   ~Pcpw 3759   U.cuni 3975   Fincfn 7068   Topctop 16913   Compccmp 17403
This theorem is referenced by:  cmpcov  17406  cncmp  17409  fincmp  17410  cmptop  17412  cmpsub  17417  tgcmp  17418  uncmp  17420  sscmp  17422  cmpfi  17425  txcmp  17628  alexsubb  18030  alexsubALT  18035  onsucsuccmpi  26097  limsucncmpi  26099  comppfsc  26277  heibor  26420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-in 3287  df-ss 3294  df-pw 3761  df-uni 3976  df-cmp 17404
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