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Theorem supwval 25284
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 5124 . . . 4  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 oveq1 5865 . . . . 5  |-  ( R  =  `' `' R  ->  ( R  sup w  A )  =  ( `' `' R  sup w  A
) )
32eqcoms 2286 . . . 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 5208 . . . 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 25245 . . . 4  |-  ( ( `' R  e.  _V  /\  A  e.  W )  ->  ( `' R  inf w  A )  =  ( `' `' R  sup w  A ) )
109eqcomd 2288 . . 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 2315 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 1623    e. wcel 1684   _Vcvv 2788   `'ccnv 4688   Rel wrel 4694  (class class class)co 5858    sup w cspw 14303    inf
w cinf 14304
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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-nfw 14309
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