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Theorem iscon 17481
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 21 . . . 4  |-  ( j  =  J  ->  j  =  J )
2 fveq2 5731 . . . 4  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
31, 2ineq12d 3545 . . 3  |-  ( j  =  J  ->  (
j  i^i  ( Clsd `  j ) )  =  ( J  i^i  ( Clsd `  J ) ) )
4 unieq 4026 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
5 iscon.1 . . . . 5  |-  X  = 
U. J
64, 5syl6eqr 2488 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
76preq2d 3892 . . 3  |-  ( j  =  J  ->  { (/) , 
U. j }  =  { (/) ,  X }
)
83, 7eqeq12d 2452 . 2  |-  ( j  =  J  ->  (
( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j }  <->  ( J  i^i  ( Clsd `  J )
)  =  { (/) ,  X } ) )
9 df-con 17480 . 2  |-  Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j } }
108, 9elrab2 3096 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321   (/)c0 3630   {cpr 3817   U.cuni 4017   ` cfv 5457   Topctop 16963   Clsdccld 17085   Conccon 17479
This theorem is referenced by:  iscon2  17482  conclo  17483  conndisj  17484  contop  17485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-con 17480
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