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Theorem ist1 19949
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 4259 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
2 ist0.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6eqr 2516 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
43eleq2d 2527 . . . 4  |-  ( x  =  J  ->  (
a  e.  U. x  <->  a  e.  X ) )
5 fveq2 5872 . . . . 5  |-  ( x  =  J  ->  ( Clsd `  x )  =  ( Clsd `  J
) )
65eleq2d 2527 . . . 4  |-  ( x  =  J  ->  ( { a }  e.  ( Clsd `  x )  <->  { a }  e.  (
Clsd `  J )
) )
74, 6imbi12d 320 . . 3  |-  ( x  =  J  ->  (
( a  e.  U. x  ->  { a }  e.  ( Clsd `  x
) )  <->  ( a  e.  X  ->  { a }  e.  ( Clsd `  J ) ) ) )
87ralbidv2 2892 . 2  |-  ( x  =  J  ->  ( A. a  e.  U. x { a }  e.  ( Clsd `  x )  <->  A. a  e.  X  {
a }  e.  (
Clsd `  J )
) )
9 df-t1 19942 . 2  |-  Fre  =  { x  e.  Top  | 
A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
108, 9elrab2 3259 1  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. a  e.  X  { a }  e.  ( Clsd `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {csn 4032   U.cuni 4251   ` cfv 5594   Topctop 19521   Clsdccld 19644   Frect1 19935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-t1 19942
This theorem is referenced by:  t1sncld  19954  t1ficld  19955  t1top  19958  ist1-2  19975  cnt1  19978  ordtt1  20007  qtopt1  27999  onint1  30098
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