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Theorem oduval 14234
Description: Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Hypotheses
Ref Expression
oduval.d  |-  D  =  (ODual `  O )
oduval.l  |-  .<_  =  ( le `  O )
Assertion
Ref Expression
oduval  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )

Proof of Theorem oduval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . 5  |-  ( a  =  O  ->  a  =  O )
2 fveq2 5525 . . . . . . 7  |-  ( a  =  O  ->  ( le `  a )  =  ( le `  O
) )
32cnveqd 4857 . . . . . 6  |-  ( a  =  O  ->  `' ( le `  a )  =  `' ( le
`  O ) )
43opeq2d 3803 . . . . 5  |-  ( a  =  O  ->  <. ( le `  ndx ) ,  `' ( le `  a ) >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >. )
51, 4oveq12d 5876 . . . 4  |-  ( a  =  O  ->  (
a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
6 df-odu 14233 . . . 4  |- ODual  =  ( a  e.  _V  |->  ( a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. ) )
7 ovex 5883 . . . 4  |-  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)  e.  _V
85, 6, 7fvmpt 5602 . . 3  |-  ( O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
9 fvprc 5519 . . . 4  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  (/) )
10 reldmsets 13170 . . . . 5  |-  Rel  dom sSet
1110ovprc1 5886 . . . 4  |-  ( -.  O  e.  _V  ->  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )  =  (/) )
129, 11eqtr4d 2318 . . 3  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
138, 12pm2.61i 156 . 2  |-  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )
14 oduval.d . 2  |-  D  =  (ODual `  O )
15 oduval.l . . . . 5  |-  .<_  =  ( le `  O )
1615cnveqi 4856 . . . 4  |-  `'  .<_  =  `' ( le `  O )
1716opeq2i 3800 . . 3  |-  <. ( le `  ndx ) ,  `'  .<_  >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >.
1817oveq2i 5869 . 2  |-  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >.
)  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)
1913, 14, 183eqtr4i 2313 1  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   lecple 13215  ODualcodu 14232
This theorem is referenced by:  oduleval  14235  odubas  14237
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
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-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sets 13154  df-odu 14233
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