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Theorem cnvps 15716
Description: The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 15717 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 5380 . . 3  |-  Rel  `' R
21a1i 11 . 2  |-  ( R  e.  PosetRel  ->  Rel  `' R
)
3 cnvco 5194 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
4 pstr2 15709 . . . 4  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
5 cnvss 5181 . . . 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 3554 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  o.  `' R )  C_  `' R )
8 psrel 15707 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
9 dfrel2 5463 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
108, 9sylib 196 . . . . 5  |-  ( R  e.  PosetRel  ->  `' `' R  =  R )
1110ineq2d 3705 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( `' R  i^i  R ) )
12 incom 3696 . . . 4  |-  ( `' R  i^i  R )  =  ( R  i^i  `' R )
1311, 12syl6eq 2524 . . 3  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( R  i^i  `' R
) )
14 psref2 15708 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
15 relcnvfld 5544 . . . . 5  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
168, 15syl 16 . . . 4  |-  ( R  e.  PosetRel  ->  U. U. R  = 
U. U. `' R )
1716reseq2d 5279 . . 3  |-  ( R  e.  PosetRel  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
1813, 14, 173eqtrd 2512 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) )
19 cnvexg 6741 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e. 
_V )
20 isps 15706 . . 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 1179 1  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   U.cuni 4251    _I cid 4796   `'ccnv 5004    |` cres 5007    o. ccom 5009   Rel wrel 5010   PosetRelcps 15702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ps 15704
This theorem is referenced by:  cnvpsb  15717  cnvtsr  15726  ordtcnv  19570  xrge0iifhmeo  27734
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