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Theorem ist1 17065
Description: The predicate  J is T1. (Contributed by FL, 18-Jun-2007.)
Hypothesis
Ref Expression
ist0.1  |-  X  = 
U. J
Assertion
Ref Expression
ist1  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J ) ) )
Distinct variable group:    J, a
Allowed substitution hint:    X( a)

Proof of Theorem ist1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 unieq 3852 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
2 ist0.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6eqr 2346 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
43eleq2d 2363 . . . 4  |-  ( x  =  J  ->  (
a  e.  U. x  <->  a  e.  X ) )
5 fveq2 5541 . . . . 5  |-  ( x  =  J  ->  ( Clsd `  x )  =  ( Clsd `  J
) )
65eleq2d 2363 . . . 4  |-  ( x  =  J  ->  ( { a }  e.  ( Clsd `  x )  <->  { a }  e.  (
Clsd `  J )
) )
74, 6imbi12d 311 . . 3  |-  ( x  =  J  ->  (
( a  e.  U. x  ->  { a }  e.  ( Clsd `  x
) )  <->  ( a  e.  X  ->  { a }  e.  ( Clsd `  J ) ) ) )
87ralbidv2 2578 . 2  |-  ( x  =  J  ->  ( A. a  e.  U. x { a }  e.  ( Clsd `  x )  <->  A. a  e.  X  {
a }  e.  (
Clsd `  J )
) )
9 df-t1 17058 . 2  |-  Fre  =  { x  e.  Top  | 
A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
108, 9elrab2 2938 1  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Frect1 17051
This theorem is referenced by:  t1sncld  17070  t1ficld  17071  t1top  17074  ist1-2  17091  cnt1  17094  ordtt1  17123  onint1  24960
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-ral 2561  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-t1 17058
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