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Theorem sege 25252
Description: The least element of a poset is the greatest element of the converse poset. (Contributed by FL, 30-Dec-2011.)
Assertion
Ref Expression
sege  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( ge
`  `' R ) )

Proof of Theorem sege
StepHypRef Expression
1 eqid 2283 . . 3  |-  dom  R  =  dom  R
21seinf 25251 . 2  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( R  inf w  dom  R
) )
3 dmexg 4939 . . 3  |-  ( R  e.  PosetRel  ->  dom  R  e.  _V )
4 nfwval 25245 . . 3  |-  ( ( R  e.  PosetRel  /\  dom  R  e.  _V )  -> 
( R  inf w  dom  R )  =  ( `' R  sup w  dom  R ) )
53, 4mpdan 649 . 2  |-  ( R  e.  PosetRel  ->  ( R  inf w  dom  R )  =  ( `' R  sup w  dom  R ) )
6 posispre 25241 . . . 4  |-  ( R  e.  PosetRel  ->  R  e. PresetRel )
71domcnvpre 25233 . . . . 5  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
87eqcomd 2288 . . . 4  |-  ( R  e. PresetRel  ->  dom  `' R  =  dom  R )
96, 8syl 15 . . 3  |-  ( R  e.  PosetRel  ->  dom  `' R  =  dom  R )
10 cnvps 14321 . . . . . 6  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
11 eqid 2283 . . . . . . 7  |-  dom  `' R  =  dom  `' R
1211gepsup 25250 . . . . . 6  |-  ( `' R  e.  PosetRel  ->  ( ge `  `' R )  =  ( `' R  sup w  dom  `' R
) )
1310, 12syl 15 . . . . 5  |-  ( R  e.  PosetRel  ->  ( ge `  `' R )  =  ( `' R  sup w  dom  `' R ) )
1413eqcomd 2288 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  sup w  dom  `' R
)  =  ( ge
`  `' R ) )
15 oveq2 5866 . . . . . 6  |-  ( dom 
R  =  dom  `' R  ->  ( `' R  sup w  dom  R )  =  ( `' R  sup w  dom  `' R
) )
1615eqcoms 2286 . . . . 5  |-  ( dom  `' R  =  dom  R  ->  ( `' R  sup w  dom  R )  =  ( `' R  sup w  dom  `' R
) )
1716eqeq1d 2291 . . . 4  |-  ( dom  `' R  =  dom  R  ->  ( ( `' R  sup w  dom  R )  =  ( ge
`  `' R )  <-> 
( `' R  sup w  dom  `' R )  =  ( ge `  `' R ) ) )
1814, 17syl5ibr 212 . . 3  |-  ( dom  `' R  =  dom  R  ->  ( R  e.  PosetRel 
->  ( `' R  sup w  dom  R )  =  ( ge `  `' R ) ) )
199, 18mpcom 32 . 2  |-  ( R  e.  PosetRel  ->  ( `' R  sup w  dom  R )  =  ( ge `  `' R ) )
202, 5, 193eqtrd 2319 1  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( ge
`  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   `'ccnv 4688   dom cdm 4689   ` cfv 5255  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303    inf w cinf 14304  PresetRelcpresetrel 25215   gecge 25220  leRcse 25221
This theorem is referenced by:  defse3  25272
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-ov 5861  df-oprab 5862  df-mpt2 5863  df-ps 14306  df-nfw 14309  df-prs 25223  df-ge 25248  df-ler 25249
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