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Theorem iscmp 18991
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 3863 . . 3  |-  ( x  =  J  ->  ~P x  =  ~P J
)
2 unieq 4099 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
3 iscmp.1 . . . . . 6  |-  X  = 
U. J
42, 3syl6eqr 2493 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
54eqeq1d 2451 . . . 4  |-  ( x  =  J  ->  ( U. x  =  U. y 
<->  X  =  U. y
) )
64eqeq1d 2451 . . . . 5  |-  ( x  =  J  ->  ( U. x  =  U. z 
<->  X  =  U. z
) )
76rexbidv 2736 . . . 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 320 . . 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 2931 . 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 18990 . 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 3119 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    i^i cin 3327   ~Pcpw 3860   U.cuni 4091   Fincfn 7310   Topctop 18498   Compccmp 18989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-in 3335  df-ss 3342  df-pw 3862  df-uni 4092  df-cmp 18990
This theorem is referenced by:  cmpcov  18992  cncmp  18995  fincmp  18996  cmptop  18998  cmpsub  19003  tgcmp  19004  uncmp  19006  sscmp  19008  cmpfi  19011  bwthOLD  19014  txcmp  19216  alexsubb  19618  alexsubALT  19623  onsucsuccmpi  28289  limsucncmpi  28291  comppfsc  28579  heibor  28720
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