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Theorem cnvps 14636
Description: The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 14637 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 5234 . . 3  |-  Rel  `' R
21a1i 11 . 2  |-  ( R  e.  PosetRel  ->  Rel  `' R
)
3 cnvco 5048 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
4 pstr2 14629 . . . 4  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
5 cnvss 5037 . . . 4  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
64, 5syl 16 . . 3  |-  ( R  e.  PosetRel  ->  `' ( R  o.  R )  C_  `' R )
73, 6syl5eqssr 3385 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  o.  `' R )  C_  `' R )
8 psrel 14627 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
9 dfrel2 5313 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
108, 9sylib 189 . . . . 5  |-  ( R  e.  PosetRel  ->  `' `' R  =  R )
1110ineq2d 3534 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( `' R  i^i  R ) )
12 incom 3525 . . . 4  |-  ( `' R  i^i  R )  =  ( R  i^i  `' R )
1311, 12syl6eq 2483 . . 3  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( R  i^i  `' R
) )
14 psref2 14628 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
15 relcnvfld 5392 . . . . 5  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
168, 15syl 16 . . . 4  |-  ( R  e.  PosetRel  ->  U. U. R  = 
U. U. `' R )
1716reseq2d 5138 . . 3  |-  ( R  e.  PosetRel  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
1813, 14, 173eqtrd 2471 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) )
19 cnvexg 5397 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e. 
_V )
20 isps 14626 . . 3  |-  ( `' R  e.  _V  ->  ( `' R  e.  PosetRel  <->  ( Rel  `' R  /\  ( `' R  o.  `' R
)  C_  `' R  /\  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) ) ) )
2119, 20syl 16 . 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 1137 1  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   U.cuni 4007    _I cid 4485   `'ccnv 4869    |` cres 4872    o. ccom 4874   Rel wrel 4875   PosetRelcps 14616
This theorem is referenced by:  cnvpsb  14637  cnvtsr  14646  ordtcnv  17257  xrge0iifhmeo  24314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ps 14621
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