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Theorem isps 14562
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 4900 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 4971 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 4972 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2420 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 id 20 . . . 4  |-  ( r  =  R  ->  r  =  R )
64, 5sseq12d 3321 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 cnveq 4987 . . . . 5  |-  ( r  =  R  ->  `' r  =  `' R
)
85, 7ineq12d 3487 . . . 4  |-  ( r  =  R  ->  (
r  i^i  `' r
)  =  ( R  i^i  `' R ) )
9 unieq 3967 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
109unieqd 3969 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
1110reseq2d 5087 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
128, 11eqeq12d 2402 . . 3  |-  ( r  =  R  ->  (
( r  i^i  `' r )  =  (  _I  |`  U. U. r
)  <->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) )
131, 6, 123anbi123d 1254 . 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 14557 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
1513, 14elab2g 3028 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 177    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3263    C_ wss 3264   U.cuni 3958    _I cid 4435   `'ccnv 4818    |` cres 4821    o. ccom 4823   Rel wrel 4824   PosetRelcps 14552
This theorem is referenced by:  psrel  14563  psref2  14564  pstr2  14565  cnvps  14572  psss  14574  letsr  14600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-v 2902  df-in 3271  df-ss 3278  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-res 4831  df-ps 14557
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