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Theorem cnvps 16458
Description: The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 16459 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
cnvps  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )

Proof of Theorem cnvps
StepHypRef Expression
1 relcnv 5207 . . 3  |-  Rel  `' R
21a1i 11 . 2  |-  ( R  e.  PosetRel  ->  Rel  `' R
)
3 cnvco 5020 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
4 pstr2 16451 . . . 4  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
5 cnvss 5007 . . . 4  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
64, 5syl 17 . . 3  |-  ( R  e.  PosetRel  ->  `' ( R  o.  R )  C_  `' R )
73, 6syl5eqssr 3477 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  o.  `' R )  C_  `' R )
8 psrel 16449 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
9 dfrel2 5286 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
108, 9sylib 200 . . . . 5  |-  ( R  e.  PosetRel  ->  `' `' R  =  R )
1110ineq2d 3634 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( `' R  i^i  R ) )
12 incom 3625 . . . 4  |-  ( `' R  i^i  R )  =  ( R  i^i  `' R )
1311, 12syl6eq 2501 . . 3  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( R  i^i  `' R
) )
14 psref2 16450 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
15 relcnvfld 5367 . . . . 5  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
168, 15syl 17 . . . 4  |-  ( R  e.  PosetRel  ->  U. U. R  = 
U. U. `' R )
1716reseq2d 5105 . . 3  |-  ( R  e.  PosetRel  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
1813, 14, 173eqtrd 2489 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) )
19 cnvexg 6739 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e. 
_V )
20 isps 16448 . . 3  |-  ( `' R  e.  _V  ->  ( `' R  e.  PosetRel  <->  ( Rel  `' R  /\  ( `' R  o.  `' R
)  C_  `' R  /\  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) ) ) )
2119, 20syl 17 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  e. 
PosetRel  <-> 
( Rel  `' R  /\  ( `' R  o.  `' R )  C_  `' R  /\  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) ) ) )
222, 7, 18, 21mpbir3and 1191 1  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045    i^i cin 3403    C_ wss 3404   U.cuni 4198    _I cid 4744   `'ccnv 4833    |` cres 4836    o. ccom 4838   Rel wrel 4839   PosetRelcps 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ps 16446
This theorem is referenced by:  cnvpsb  16459  cnvtsr  16468  ordtcnv  20217  xrge0iifhmeo  28742
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