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Theorem oduval 14250
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 5541 . . . . . . 7  |-  ( a  =  O  ->  ( le `  a )  =  ( le `  O
) )
32cnveqd 4873 . . . . . 6  |-  ( a  =  O  ->  `' ( le `  a )  =  `' ( le
`  O ) )
43opeq2d 3819 . . . . 5  |-  ( a  =  O  ->  <. ( le `  ndx ) ,  `' ( le `  a ) >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >. )
51, 4oveq12d 5892 . . . 4  |-  ( a  =  O  ->  (
a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
6 df-odu 14249 . . . 4  |- ODual  =  ( a  e.  _V  |->  ( a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. ) )
7 ovex 5899 . . . 4  |-  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)  e.  _V
85, 6, 7fvmpt 5618 . . 3  |-  ( O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
9 fvprc 5535 . . . 4  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  (/) )
10 reldmsets 13186 . . . . 5  |-  Rel  dom sSet
1110ovprc1 5902 . . . 4  |-  ( -.  O  e.  _V  ->  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )  =  (/) )
129, 11eqtr4d 2331 . . 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 4872 . . . 4  |-  `'  .<_  =  `' ( le `  O )
1716opeq2i 3816 . . 3  |-  <. ( le `  ndx ) ,  `'  .<_  >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >.
1817oveq2i 5885 . 2  |-  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >.
)  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)
1913, 14, 183eqtr4i 2326 1  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   ndxcnx 13161   sSet csts 13162   lecple 13231  ODualcodu 14248
This theorem is referenced by:  oduleval  14251  odubas  14253
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
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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-sets 13170  df-odu 14249
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