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Theorem isps 16031
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 5073 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 5149 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 5150 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2495 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 id 22 . . . 4  |-  ( r  =  R  ->  r  =  R )
64, 5sseq12d 3518 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 cnveq 5165 . . . . 5  |-  ( r  =  R  ->  `' r  =  `' R
)
85, 7ineq12d 3687 . . . 4  |-  ( r  =  R  ->  (
r  i^i  `' r
)  =  ( R  i^i  `' R ) )
9 unieq 4243 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
109unieqd 4245 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
1110reseq2d 5262 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
128, 11eqeq12d 2476 . . 3  |-  ( r  =  R  ->  (
( r  i^i  `' r )  =  (  _I  |`  U. U. r
)  <->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) )
131, 6, 123anbi123d 1297 . 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 16029 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
1513, 14elab2g 3245 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   U.cuni 4235    _I cid 4779   `'ccnv 4987    |` cres 4990    o. ccom 4992   Rel wrel 4993   PosetRelcps 16027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-res 5000  df-ps 16029
This theorem is referenced by:  psrel  16032  psref2  16033  pstr2  16034  cnvps  16041  psss  16043  letsr  16056
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