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Theorem mnlmxl2 25372
Description: The minimal elements of a preset are the maximal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
mnlmxl2  |-  ( R  e. PresetRel  ->  ( mnl `  R
)  =  ( mxl `  `' R ) )

Proof of Theorem mnlmxl2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . 4  |-  dom  R  =  dom  R
21domcnvpre 25336 . . 3  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
3 vex 2804 . . . . . . 7  |-  x  e. 
_V
4 vex 2804 . . . . . . 7  |-  y  e. 
_V
5 brcnvg 4878 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x `' R
y  <->  y R x ) )
65bicomd 192 . . . . . . 7  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( y R x  <-> 
x `' R y ) )
73, 4, 6mp2an 653 . . . . . 6  |-  ( y R x  <->  x `' R y )
8 equcom 1665 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
97, 8imbi12i 316 . . . . 5  |-  ( ( y R x  -> 
y  =  x )  <-> 
( x `' R
y  ->  x  =  y ) )
109a1i 10 . . . 4  |-  ( R  e. PresetRel  ->  ( ( y R x  ->  y  =  x )  <->  ( x `' R y  ->  x  =  y ) ) )
112, 10raleqbidv 2761 . . 3  |-  ( R  e. PresetRel  ->  ( A. y  e.  dom  R ( y R x  ->  y  =  x )  <->  A. y  e.  dom  `' R ( x `' R y  ->  x  =  y ) ) )
122, 11rabeqbidv 2796 . 2  |-  ( R  e. PresetRel  ->  { x  e. 
dom  R  |  A. y  e.  dom  R ( y R x  -> 
y  =  x ) }  =  { x  e.  dom  `' R  |  A. y  e.  dom  `' R ( x `' R y  ->  x  =  y ) } )
131mnlelt2 25369 . 2  |-  ( R  e. PresetRel  ->  ( mnl `  R
)  =  { x  e.  dom  R  |  A. y  e.  dom  R ( y R x  -> 
y  =  x ) } )
14 dupre1 25346 . . 3  |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
15 eqid 2296 . . . 4  |-  dom  `' R  =  dom  `' R
1615mxlelt2 25368 . . 3  |-  ( `' R  e. PresetRel  ->  ( mxl `  `' R )  =  {
x  e.  dom  `' R  |  A. y  e.  dom  `' R ( x `' R y  ->  x  =  y ) } )
1714, 16syl 15 . 2  |-  ( R  e. PresetRel  ->  ( mxl `  `' R )  =  {
x  e.  dom  `' R  |  A. y  e.  dom  `' R ( x `' R y  ->  x  =  y ) } )
1812, 13, 173eqtr4d 2338 1  |-  ( R  e. PresetRel  ->  ( mnl `  R
)  =  ( mxl `  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ` cfv 5271  PresetRelcpresetrel 25318   mxlcmxl 25319   mnlcmnl 25320
This theorem is referenced by:  mxlmnl2  25373
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-prs 25326  df-mxl 25349  df-mnl 25350
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