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Theorem iscmp 17366
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 3738 . . 3  |-  ( x  =  J  ->  ~P x  =  ~P J
)
2 unieq 3959 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
3 iscmp.1 . . . . . 6  |-  X  = 
U. J
42, 3syl6eqr 2430 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
54eqeq1d 2388 . . . 4  |-  ( x  =  J  ->  ( U. x  =  U. y 
<->  X  =  U. y
) )
64eqeq1d 2388 . . . . 5  |-  ( x  =  J  ->  ( U. x  =  U. z 
<->  X  =  U. z
) )
76rexbidv 2663 . . . 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 2852 . 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 17365 . 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 3030 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 1717   A.wral 2642   E.wrex 2643    i^i cin 3255   ~Pcpw 3735   U.cuni 3950   Fincfn 7038   Topctop 16874   Compccmp 17364
This theorem is referenced by:  cmpcov  17367  cncmp  17370  fincmp  17371  cmptop  17373  cmpsub  17378  tgcmp  17379  uncmp  17381  sscmp  17383  cmpfi  17386  txcmp  17589  alexsubb  17991  alexsubALT  17996  onsucsuccmpi  25900  limsucncmpi  25902  comppfsc  26071  heibor  26214
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-in 3263  df-ss 3270  df-pw 3737  df-uni 3951  df-cmp 17365
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