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Theorem sege 25355
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 2296 . . 3  |-  dom  R  =  dom  R
21seinf 25354 . 2  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( R  inf w  dom  R
) )
3 dmexg 4955 . . 3  |-  ( R  e.  PosetRel  ->  dom  R  e.  _V )
4 nfwval 25348 . . 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 25344 . . . 4  |-  ( R  e.  PosetRel  ->  R  e. PresetRel )
71domcnvpre 25336 . . . . 5  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
87eqcomd 2301 . . . 4  |-  ( R  e. PresetRel  ->  dom  `' R  =  dom  R )
96, 8syl 15 . . 3  |-  ( R  e.  PosetRel  ->  dom  `' R  =  dom  R )
10 cnvps 14337 . . . . . 6  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
11 eqid 2296 . . . . . . 7  |-  dom  `' R  =  dom  `' R
1211gepsup 25353 . . . . . 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 2301 . . . 4  |-  ( R  e.  PosetRel  ->  ( `' R  sup w  dom  `' R
)  =  ( ge
`  `' R ) )
15 oveq2 5882 . . . . . 6  |-  ( dom 
R  =  dom  `' R  ->  ( `' R  sup w  dom  R )  =  ( `' R  sup w  dom  `' R
) )
1615eqcoms 2299 . . . . 5  |-  ( dom  `' R  =  dom  R  ->  ( `' R  sup w  dom  R )  =  ( `' R  sup w  dom  `' R
) )
1716eqeq1d 2304 . . . 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 2332 1  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( ge
`  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   `'ccnv 4704   dom cdm 4705   ` cfv 5271  (class class class)co 5874   PosetRelcps 14317    sup w cspw 14319    inf w cinf 14320  PresetRelcpresetrel 25318   gecge 25323  leRcse 25324
This theorem is referenced by:  defse3  25375
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-ov 5877  df-oprab 5878  df-mpt2 5879  df-ps 14322  df-nfw 14325  df-prs 25326  df-ge 25351  df-ler 25352
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