Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isprsr Unicode version

Theorem isprsr 25327
Description: The predicate "is a preset". (Contributed by FL, 1-May-2011.)
Assertion
Ref Expression
isprsr  |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )

Proof of Theorem isprsr
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 releq 4787 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 4857 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 4858 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2328 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 sseq12 3214 . . . 4  |-  ( ( ( r  o.  r
)  =  ( R  o.  R )  /\  r  =  R )  ->  ( ( r  o.  r )  C_  r  <->  ( R  o.  R ) 
C_  R ) )
64, 5mpancom 650 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 unieq 3852 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
87unieqd 3854 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
98reseq2d 4971 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
10 sseq12 3214 . . . 4  |-  ( ( (  _I  |`  U. U. r )  =  (  _I  |`  U. U. R
)  /\  r  =  R )  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  U. U. R
)  C_  R )
)
119, 10mpancom 650 . . 3  |-  ( r  =  R  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  U. U. R
)  C_  R )
)
121, 6, 113anbi123d 1252 . 2  |-  ( r  =  R  ->  (
( Rel  r  /\  ( r  o.  r
)  C_  r  /\  (  _I  |`  U. U. r )  C_  r
)  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
13 df-prs 25326 . 2  |- PresetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) }
1412, 13elab2g 2929 1  |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843    _I cid 4320    |` cres 4707    o. ccom 4709   Rel wrel 4710  PresetRelcpresetrel 25318
This theorem is referenced by:  preorel  25328  preodom2  25329  preoref12  25331  preoran2  25333  preotr1  25337  altprs2  25339  int2pre  25340  sqpre  25341  dupre1  25346  dfdir2  25394
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-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-co 4714  df-res 4717  df-prs 25326
  Copyright terms: Public domain W3C validator