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Theorem ist1 17300
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 3959 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
2 ist0.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6eqr 2430 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
43eleq2d 2447 . . . 4  |-  ( x  =  J  ->  (
a  e.  U. x  <->  a  e.  X ) )
5 fveq2 5661 . . . . 5  |-  ( x  =  J  ->  ( Clsd `  x )  =  ( Clsd `  J
) )
65eleq2d 2447 . . . 4  |-  ( x  =  J  ->  ( { a }  e.  ( Clsd `  x )  <->  { a }  e.  (
Clsd `  J )
) )
74, 6imbi12d 312 . . 3  |-  ( x  =  J  ->  (
( a  e.  U. x  ->  { a }  e.  ( Clsd `  x
) )  <->  ( a  e.  X  ->  { a }  e.  ( Clsd `  J ) ) ) )
87ralbidv2 2664 . 2  |-  ( x  =  J  ->  ( A. a  e.  U. x { a }  e.  ( Clsd `  x )  <->  A. a  e.  X  {
a }  e.  (
Clsd `  J )
) )
9 df-t1 17293 . 2  |-  Fre  =  { x  e.  Top  | 
A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
108, 9elrab2 3030 1  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {csn 3750   U.cuni 3950   ` cfv 5387   Topctop 16874   Clsdccld 16996   Frect1 17286
This theorem is referenced by:  t1sncld  17305  t1ficld  17306  t1top  17309  ist1-2  17326  cnt1  17329  ordtt1  17358  onint1  25906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-t1 17293
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