MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscon Structured version   Unicode version

Theorem iscon 19022
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 5696 . . . 4  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
31, 2ineq12d 3558 . . 3  |-  ( j  =  J  ->  (
j  i^i  ( Clsd `  j ) )  =  ( J  i^i  ( Clsd `  J ) ) )
4 unieq 4104 . . . . 5  |-  ( j  =  J  ->  U. j  =  U. J )
5 iscon.1 . . . . 5  |-  X  = 
U. J
64, 5syl6eqr 2493 . . . 4  |-  ( j  =  J  ->  U. j  =  X )
76preq2d 3966 . . 3  |-  ( j  =  J  ->  { (/) , 
U. j }  =  { (/) ,  X }
)
83, 7eqeq12d 2457 . 2  |-  ( j  =  J  ->  (
( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j }  <->  ( J  i^i  ( Clsd `  J )
)  =  { (/) ,  X } ) )
9 df-con 19021 . 2  |-  Con  =  { j  e.  Top  |  ( j  i^i  ( Clsd `  j ) )  =  { (/) ,  U. j } }
108, 9elrab2 3124 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3332   (/)c0 3642   {cpr 3884   U.cuni 4096   ` cfv 5423   Topctop 18503   Clsdccld 18625   Conccon 19020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-con 19021
This theorem is referenced by:  iscon2  19023  conclo  19024  conndisj  19025  contop  19026
  Copyright terms: Public domain W3C validator