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Theorem iscon 17155
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 5541 . . . 4  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
31, 2ineq12d 3384 . . 3  |-  ( j  =  J  ->  (
j  i^i  ( Clsd `  j ) )  =  ( J  i^i  ( Clsd `  J ) ) )
4 unieq 3852 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
5 iscon.1 . . . . 5  |-  X  = 
U. J
64, 5syl6eqr 2346 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
76preq2d 3726 . . 3  |-  ( j  =  J  ->  { (/) , 
U. j }  =  { (/) ,  X }
)
83, 7eqeq12d 2310 . 2  |-  ( j  =  J  ->  (
( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j }  <->  ( J  i^i  ( Clsd `  J )
)  =  { (/) ,  X } ) )
9 df-con 17154 . 2  |-  Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j } }
108, 9elrab2 2938 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 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  iscon2  17156  conclo  17157  conndisj  17158  contop  17159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-con 17154
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