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Theorem iscmp 19647
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 4006 . . 3  |-  ( x  =  J  ->  ~P x  =  ~P J
)
2 unieq 4246 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
3 iscmp.1 . . . . . 6  |-  X  = 
U. J
42, 3syl6eqr 2519 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
54eqeq1d 2462 . . . 4  |-  ( x  =  J  ->  ( U. x  =  U. y 
<->  X  =  U. y
) )
64eqeq1d 2462 . . . . 5  |-  ( x  =  J  ->  ( U. x  =  U. z 
<->  X  =  U. z
) )
76rexbidv 2966 . . . 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 3065 . 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 19646 . 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 3256 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 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    i^i cin 3468   ~Pcpw 4003   U.cuni 4238   Fincfn 7506   Topctop 19154   Compccmp 19645
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 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-in 3476  df-ss 3483  df-pw 4005  df-uni 4239  df-cmp 19646
This theorem is referenced by:  cmpcov  19648  cncmp  19651  fincmp  19652  cmptop  19654  cmpsub  19659  tgcmp  19660  uncmp  19662  sscmp  19664  cmpfi  19667  bwthOLD  19670  txcmp  19872  alexsubb  20274  alexsubALT  20279  onsucsuccmpi  29471  limsucncmpi  29473  comppfsc  29766  heibor  29907
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