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Theorem mnlmxl2 24681
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 2283 . . . 4  |-  dom  R  =  dom  R
21domcnvpre 24645 . . 3  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
3 vex 2791 . . . . . . 7  |-  x  e. 
_V
4 vex 2791 . . . . . . 7  |-  y  e. 
_V
5 brcnvg 4862 . . . . . . . 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 1647 . . . . . 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 2748 . . 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 2783 . 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 24678 . 2  |-  ( R  e. PresetRel  ->  ( mnl `  R
)  =  { x  e.  dom  R  |  A. y  e.  dom  R ( y R x  -> 
y  =  x ) } )
14 dupre1 24655 . . 3  |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
15 eqid 2283 . . . 4  |-  dom  `' R  =  dom  `' R
1615mxlelt2 24677 . . 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 2325 1  |-  ( R  e. PresetRel  ->  ( mnl `  R
)  =  ( mxl `  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ` cfv 5255  PresetRelcpresetrel 24627   mxlcmxl 24628   mnlcmnl 24629
This theorem is referenced by:  mxlmnl2  24682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-prs 24635  df-mxl 24658  df-mnl 24659
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