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Theorem iscon 20363
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 22 . . . 4  |-  ( j  =  J  ->  j  =  J )
2 fveq2 5818 . . . 4  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
31, 2ineq12d 3601 . . 3  |-  ( j  =  J  ->  (
j  i^i  ( Clsd `  j ) )  =  ( J  i^i  ( Clsd `  J ) ) )
4 unieq 4163 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
5 iscon.1 . . . . 5  |-  X  = 
U. J
64, 5syl6eqr 2474 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
76preq2d 4022 . . 3  |-  ( j  =  J  ->  { (/) , 
U. j }  =  { (/) ,  X }
)
83, 7eqeq12d 2437 . 2  |-  ( j  =  J  ->  (
( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j }  <->  ( J  i^i  ( Clsd `  J )
)  =  { (/) ,  X } ) )
9 df-con 20362 . 2  |-  Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j } }
108, 9elrab2 3166 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    i^i cin 3371   (/)c0 3697   {cpr 3936   U.cuni 4155   ` cfv 5537   Topctop 19852   Clsdccld 19966   Conccon 20361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-rex 2714  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-iota 5501  df-fv 5545  df-con 20362
This theorem is referenced by:  iscon2  20364  conclo  20365  conndisj  20366  contop  20367
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