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Theorem isps 15483
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 5023 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 5098 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 5099 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2492 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 id 22 . . . 4  |-  ( r  =  R  ->  r  =  R )
64, 5sseq12d 3486 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 cnveq 5114 . . . . 5  |-  ( r  =  R  ->  `' r  =  `' R
)
85, 7ineq12d 3654 . . . 4  |-  ( r  =  R  ->  (
r  i^i  `' r
)  =  ( R  i^i  `' R ) )
9 unieq 4200 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
109unieqd 4202 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
1110reseq2d 5211 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
128, 11eqeq12d 2473 . . 3  |-  ( r  =  R  ->  (
( r  i^i  `' r )  =  (  _I  |`  U. U. r
)  <->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) ) )
131, 6, 123anbi123d 1290 . 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 15481 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
1513, 14elab2g 3208 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 965    = wceq 1370    e. wcel 1758    i^i cin 3428    C_ wss 3429   U.cuni 4192    _I cid 4732   `'ccnv 4940    |` cres 4943    o. ccom 4945   Rel wrel 4946   PosetRelcps 15479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-v 3073  df-in 3436  df-ss 3443  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-res 4953  df-ps 15481
This theorem is referenced by:  psrel  15484  psref2  15485  pstr2  15486  cnvps  15493  psss  15495  letsr  15508
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