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Theorem iscon 17139
Description: The predicate  J is a connected topology . (Contributed by FL, 17-Nov-2008.)
Hypothesis
Ref Expression
iscon.1  |-  X  = 
U. J
Assertion
Ref Expression
iscon  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )

Proof of Theorem iscon
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( j  =  J  ->  j  =  J )
2 fveq2 5525 . . . 4  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
31, 2ineq12d 3371 . . 3  |-  ( j  =  J  ->  (
j  i^i  ( Clsd `  j ) )  =  ( J  i^i  ( Clsd `  J ) ) )
4 unieq 3836 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
5 iscon.1 . . . . 5  |-  X  = 
U. J
64, 5syl6eqr 2333 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
76preq2d 3713 . . 3  |-  ( j  =  J  ->  { (/) , 
U. j }  =  { (/) ,  X }
)
83, 7eqeq12d 2297 . 2  |-  ( j  =  J  ->  (
( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j }  <->  ( J  i^i  ( Clsd `  J )
)  =  { (/) ,  X } ) )
9 df-con 17138 . 2  |-  Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j } }
108, 9elrab2 2925 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   (/)c0 3455   {cpr 3641   U.cuni 3827   ` cfv 5255   Topctop 16631   Clsdccld 16753   Conccon 17137
This theorem is referenced by:  iscon2  17140  conclo  17141  conndisj  17142  contop  17143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-con 17138
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