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Theorem isps 14311
Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
Assertion
Ref Expression
isps  |-  ( R  e.  A  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  ( R  i^i  `' R
)  =  (  _I  |`  U. U. R ) ) ) )

Proof of Theorem isps
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 releq 4771 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 4841 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 4842 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2315 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 id 19 . . . 4  |-  ( r  =  R  ->  r  =  R )
64, 5sseq12d 3207 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 cnveq 4855 . . . . 5  |-  ( r  =  R  ->  `' r  =  `' R
)
85, 7ineq12d 3371 . . . 4  |-  ( r  =  R  ->  (
r  i^i  `' r
)  =  ( R  i^i  `' R ) )
9 unieq 3836 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
109unieqd 3838 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
1110reseq2d 4955 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
128, 11eqeq12d 2297 . . 3  |-  ( r  =  R  ->  (
( r  i^i  `' r )  =  (  _I  |`  U. U. r
)  <->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) )
131, 6, 123anbi123d 1252 . 2  |-  ( r  =  R  ->  (
( Rel  r  /\  ( r  o.  r
)  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r
) )  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) ) )
14 df-ps 14306 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
1513, 14elab2g 2916 1  |-  ( R  e.  A  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  ( R  i^i  `' R
)  =  (  _I  |`  U. U. R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ 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:  psrel  14312  psref2  14313  pstr2  14314  cnvps  14321  psss  14323  letsr  14349  empos  25242  inposet  25278  dispos  25287
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
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