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Theorem cnvps 15504
Description: The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 15505 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 5317 . . 3  |-  Rel  `' R
21a1i 11 . 2  |-  ( R  e.  PosetRel  ->  Rel  `' R
)
3 cnvco 5136 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
4 pstr2 15497 . . . 4  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
5 cnvss 5123 . . . 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 3512 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  o.  `' R )  C_  `' R )
8 psrel 15495 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
9 dfrel2 5399 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
108, 9sylib 196 . . . . 5  |-  ( R  e.  PosetRel  ->  `' `' R  =  R )
1110ineq2d 3663 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( `' R  i^i  R ) )
12 incom 3654 . . . 4  |-  ( `' R  i^i  R )  =  ( R  i^i  `' R )
1311, 12syl6eq 2511 . . 3  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( R  i^i  `' R
) )
14 psref2 15496 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
15 relcnvfld 5479 . . . . 5  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
168, 15syl 16 . . . 4  |-  ( R  e.  PosetRel  ->  U. U. R  = 
U. U. `' R )
1716reseq2d 5221 . . 3  |-  ( R  e.  PosetRel  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
1813, 14, 173eqtrd 2499 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) )
19 cnvexg 6637 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e. 
_V )
20 isps 15494 . . 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 1171 1  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   U.cuni 4202    _I cid 4742   `'ccnv 4950    |` cres 4953    o. ccom 4955   Rel wrel 4956   PosetRelcps 15490
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ps 15492
This theorem is referenced by:  cnvpsb  15505  cnvtsr  15514  ordtcnv  18940  xrge0iifhmeo  26531
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