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Theorem cnvps 14337
Description: The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 14338 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 5067 . . 3  |-  Rel  `' R
21a1i 10 . 2  |-  ( R  e.  PosetRel  ->  Rel  `' R
)
3 cnvco 4881 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
4 pstr2 14330 . . . 4  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
5 cnvss 4870 . . . 4  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
64, 5syl 15 . . 3  |-  ( R  e.  PosetRel  ->  `' ( R  o.  R )  C_  `' R )
73, 6syl5eqssr 3236 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  o.  `' R )  C_  `' R )
8 psrel 14328 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
9 dfrel2 5140 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
108, 9sylib 188 . . . . 5  |-  ( R  e.  PosetRel  ->  `' `' R  =  R )
1110ineq2d 3383 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( `' R  i^i  R ) )
12 incom 3374 . . . 4  |-  ( `' R  i^i  R )  =  ( R  i^i  `' R )
1311, 12syl6eq 2344 . . 3  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  ( R  i^i  `' R
) )
14 psref2 14329 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
15 relcnvfld 5219 . . . . 5  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
168, 15syl 15 . . . 4  |-  ( R  e.  PosetRel  ->  U. U. R  = 
U. U. `' R )
1716reseq2d 4971 . . 3  |-  ( R  e.  PosetRel  ->  (  _I  |`  U. U. R )  =  (  _I  |`  U. U. `' R ) )
1813, 14, 173eqtrd 2332 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) )
19 cnvexg 5224 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e. 
_V )
20 isps 14327 . . 3  |-  ( `' R  e.  _V  ->  ( `' R  e.  PosetRel  <->  ( Rel  `' R  /\  ( `' R  o.  `' R
)  C_  `' R  /\  ( `' R  i^i  `' `' R )  =  (  _I  |`  U. U. `' R ) ) ) )
2119, 20syl 15 . 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 1135 1  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   U.cuni 3843    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   Rel wrel 4710   PosetRelcps 14317
This theorem is referenced by:  cnvpsb  14338  cnvtsr  14347  ordtcnv  16947  xrge0iifhmeo  23333  sege  25355  defse3  25375  nfwpr4c  25388  toplat  25393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ps 14322
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