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Theorem oduval 15296
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 22 . . . . 5  |-  ( a  =  O  ->  a  =  O )
2 fveq2 5688 . . . . . . 7  |-  ( a  =  O  ->  ( le `  a )  =  ( le `  O
) )
32cnveqd 5011 . . . . . 6  |-  ( a  =  O  ->  `' ( le `  a )  =  `' ( le
`  O ) )
43opeq2d 4063 . . . . 5  |-  ( a  =  O  ->  <. ( le `  ndx ) ,  `' ( le `  a ) >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >. )
51, 4oveq12d 6108 . . . 4  |-  ( a  =  O  ->  (
a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
6 df-odu 15295 . . . 4  |- ODual  =  ( a  e.  _V  |->  ( a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. ) )
7 ovex 6115 . . . 4  |-  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)  e.  _V
85, 6, 7fvmpt 5771 . . 3  |-  ( O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
9 fvprc 5682 . . . 4  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  (/) )
10 reldmsets 14192 . . . . 5  |-  Rel  dom sSet
1110ovprc1 6118 . . . 4  |-  ( -.  O  e.  _V  ->  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )  =  (/) )
129, 11eqtr4d 2476 . . 3  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
138, 12pm2.61i 164 . 2  |-  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )
14 oduval.d . 2  |-  D  =  (ODual `  O )
15 oduval.l . . . . 5  |-  .<_  =  ( le `  O )
1615cnveqi 5010 . . . 4  |-  `'  .<_  =  `' ( le `  O )
1716opeq2i 4060 . . 3  |-  <. ( le `  ndx ) ,  `'  .<_  >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >.
1817oveq2i 6101 . 2  |-  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >.
)  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)
1913, 14, 183eqtr4i 2471 1  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1364    e. wcel 1761   _Vcvv 2970   (/)c0 3634   <.cop 3880   `'ccnv 4835   ` cfv 5415  (class class class)co 6090   ndxcnx 14167   sSet csts 14168   lecple 14241  ODualcodu 15294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-sets 14176  df-odu 15295
This theorem is referenced by:  oduleval  15297  odubas  15299
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