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Theorem isps 16459
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 4895 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 4970 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 4971 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2486 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 id 22 . . . 4  |-  ( r  =  R  ->  r  =  R )
64, 5sseq12d 3429 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 cnveq 4986 . . . . 5  |-  ( r  =  R  ->  `' r  =  `' R
)
85, 7ineq12d 3603 . . . 4  |-  ( r  =  R  ->  (
r  i^i  `' r
)  =  ( R  i^i  `' R ) )
9 unieq 4176 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
109unieqd 4178 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
1110reseq2d 5083 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
128, 11eqeq12d 2467 . . 3  |-  ( r  =  R  ->  (
( r  i^i  `' r )  =  (  _I  |`  U. U. r
)  <->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) )
131, 6, 123anbi123d 1343 . 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 16457 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
1513, 14elab2g 3155 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 189    /\ w3a 986    = wceq 1448    e. wcel 1891    i^i cin 3371    C_ wss 3372   U.cuni 4168    _I cid 4722   `'ccnv 4811    |` cres 4814    o. ccom 4816   Rel wrel 4817   PosetRelcps 16455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-rex 2743  df-v 3015  df-in 3379  df-ss 3386  df-uni 4169  df-br 4375  df-opab 4434  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-res 4824  df-ps 16457
This theorem is referenced by:  psrel  16460  psref2  16461  pstr2  16462  cnvps  16469  psss  16471  letsr  16484
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