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Theorem ist1 17377
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 4016 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
2 ist0.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6eqr 2485 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
43eleq2d 2502 . . . 4  |-  ( x  =  J  ->  (
a  e.  U. x  <->  a  e.  X ) )
5 fveq2 5720 . . . . 5  |-  ( x  =  J  ->  ( Clsd `  x )  =  ( Clsd `  J
) )
65eleq2d 2502 . . . 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 2719 . 2  |-  ( x  =  J  ->  ( A. a  e.  U. x { a }  e.  ( Clsd `  x )  <->  A. a  e.  X  {
a }  e.  (
Clsd `  J )
) )
9 df-t1 17370 . 2  |-  Fre  =  { x  e.  Top  | 
A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
108, 9elrab2 3086 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 1652    e. wcel 1725   A.wral 2697   {csn 3806   U.cuni 4007   ` cfv 5446   Topctop 16950   Clsdccld 17072   Frect1 17363
This theorem is referenced by:  t1sncld  17382  t1ficld  17383  t1top  17386  ist1-2  17403  cnt1  17406  ordtt1  17435  onint1  26191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-t1 17370
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