Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  supwval Unicode version

Theorem supwval 25387
Description: Value of an infimum under a weak ordering. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
supwval  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( R  sup w  A )  =  ( `' R  inf w  A ) )

Proof of Theorem supwval
StepHypRef Expression
1 dfrel2 5140 . . . 4  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 oveq1 5881 . . . . 5  |-  ( R  =  `' `' R  ->  ( R  sup w  A )  =  ( `' `' R  sup w  A
) )
32eqcoms 2299 . . . 4  |-  ( `' `' R  =  R  ->  ( R  sup w  A )  =  ( `' `' R  sup w  A
) )
41, 3sylbi 187 . . 3  |-  ( Rel 
R  ->  ( R  sup w  A )  =  ( `' `' R  sup w  A ) )
543ad2ant1 976 . 2  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( R  sup w  A )  =  ( `' `' R  sup w  A ) )
6 cnvexg 5224 . . . 4  |-  ( R  e.  U  ->  `' R  e.  _V )
763ad2ant2 977 . . 3  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  `' R  e.  _V )
8 simp3 957 . . 3  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  A  e.  W )
9 nfwval 25348 . . . 4  |-  ( ( `' R  e.  _V  /\  A  e.  W )  ->  ( `' R  inf w  A )  =  ( `' `' R  sup w  A ) )
109eqcomd 2301 . . 3  |-  ( ( `' R  e.  _V  /\  A  e.  W )  ->  ( `' `' R  sup w  A )  =  ( `' R  inf w  A ) )
117, 8, 10syl2anc 642 . 2  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( `' `' R  sup w  A
)  =  ( `' R  inf w  A
) )
125, 11eqtrd 2328 1  |-  ( ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( R  sup w  A )  =  ( `' R  inf w  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   `'ccnv 4704   Rel wrel 4710  (class class class)co 5874    sup w cspw 14319    inf
w cinf 14320
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-rab 2565  df-v 2803  df-sbc 3005  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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-nfw 14325
  Copyright terms: Public domain W3C validator