MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psref2 Unicode version

Theorem psref2 14313
Description: A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
psref2  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )

Proof of Theorem psref2
StepHypRef Expression
1 isps 14311 . . 3  |-  ( R  e.  PosetRel  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) ) ) )
21ibi 232 . 2  |-  ( R  e.  PosetRel  ->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  ( R  i^i  `' R
)  =  (  _I  |`  U. U. R ) ) )
32simp3d 969 1  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694   PosetRelcps 14301
This theorem is referenced by:  pslem  14315  cnvps  14321  tsrdir  14360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-ps 14306
  Copyright terms: Public domain W3C validator