MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ist1 Structured version   Unicode version

Theorem ist1 18925
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 4099 . . . . . 6  |-  ( x  =  J  ->  U. x  =  U. J )
2 ist0.1 . . . . . 6  |-  X  = 
U. J
31, 2syl6eqr 2493 . . . . 5  |-  ( x  =  J  ->  U. x  =  X )
43eleq2d 2510 . . . 4  |-  ( x  =  J  ->  (
a  e.  U. x  <->  a  e.  X ) )
5 fveq2 5691 . . . . 5  |-  ( x  =  J  ->  ( Clsd `  x )  =  ( Clsd `  J
) )
65eleq2d 2510 . . . 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 2737 . 2  |-  ( x  =  J  ->  ( A. a  e.  U. x { a }  e.  ( Clsd `  x )  <->  A. a  e.  X  {
a }  e.  (
Clsd `  J )
) )
9 df-t1 18918 . 2  |-  Fre  =  { x  e.  Top  | 
A. a  e.  U. x { a }  e.  ( Clsd `  x ) }
108, 9elrab2 3119 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 1369    e. wcel 1756   A.wral 2715   {csn 3877   U.cuni 4091   ` cfv 5418   Topctop 18498   Clsdccld 18620   Frect1 18911
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 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-t1 18918
This theorem is referenced by:  t1sncld  18930  t1ficld  18931  t1top  18934  ist1-2  18951  cnt1  18954  ordtt1  18983  onint1  28295
  Copyright terms: Public domain W3C validator