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Theorem mxlmnl2 25270
Description: The maximal elements of a preset are the minimal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
mxlmnl2  |-  ( R  e. PresetRel  ->  ( mxl `  R
)  =  ( mnl `  `' R ) )

Proof of Theorem mxlmnl2
StepHypRef Expression
1 preorel 25225 . . 3  |-  ( R  e. PresetRel  ->  Rel  R )
2 dfrel2 5124 . . 3  |-  ( Rel 
R  <->  `' `' R  =  R
)
31, 2sylib 188 . 2  |-  ( R  e. PresetRel  ->  `' `' R  =  R )
4 dupre1 25243 . . . . . 6  |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
5 mnlmxl2 25269 . . . . . 6  |-  ( `' R  e. PresetRel  ->  ( mnl `  `' R )  =  ( mxl `  `' `' R ) )
64, 5syl 15 . . . . 5  |-  ( R  e. PresetRel  ->  ( mnl `  `' R )  =  ( mxl `  `' `' R ) )
76eqcomd 2288 . . . 4  |-  ( R  e. PresetRel  ->  ( mxl `  `' `' R )  =  ( mnl `  `' R
) )
8 fveq2 5525 . . . . 5  |-  ( R  =  `' `' R  ->  ( mxl `  R
)  =  ( mxl `  `' `' R ) )
98eqeq1d 2291 . . . 4  |-  ( R  =  `' `' R  ->  ( ( mxl `  R
)  =  ( mnl `  `' R )  <->  ( mxl `  `' `' R )  =  ( mnl `  `' R
) ) )
107, 9syl5ibr 212 . . 3  |-  ( R  =  `' `' R  ->  ( R  e. PresetRel  ->  ( mxl `  R )  =  ( mnl `  `' R ) ) )
1110eqcoms 2286 . 2  |-  ( `' `' R  =  R  ->  ( R  e. PresetRel  ->  ( mxl `  R )  =  ( mnl `  `' R ) ) )
123, 11mpcom 32 1  |-  ( R  e. PresetRel  ->  ( mxl `  R
)  =  ( mnl `  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   `'ccnv 4688   Rel wrel 4694   ` cfv 5255  PresetRelcpresetrel 25215   mxlcmxl 25216   mnlcmnl 25217
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 25223  df-mxl 25246  df-mnl 25247
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