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Theorem oduval 14512
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 20 . . . . 5  |-  ( a  =  O  ->  a  =  O )
2 fveq2 5687 . . . . . . 7  |-  ( a  =  O  ->  ( le `  a )  =  ( le `  O
) )
32cnveqd 5007 . . . . . 6  |-  ( a  =  O  ->  `' ( le `  a )  =  `' ( le
`  O ) )
43opeq2d 3951 . . . . 5  |-  ( a  =  O  ->  <. ( le `  ndx ) ,  `' ( le `  a ) >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >. )
51, 4oveq12d 6058 . . . 4  |-  ( a  =  O  ->  (
a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
6 df-odu 14511 . . . 4  |- ODual  =  ( a  e.  _V  |->  ( a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. ) )
7 ovex 6065 . . . 4  |-  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)  e.  _V
85, 6, 7fvmpt 5765 . . 3  |-  ( O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
9 fvprc 5681 . . . 4  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  (/) )
10 reldmsets 13446 . . . . 5  |-  Rel  dom sSet
1110ovprc1 6068 . . . 4  |-  ( -.  O  e.  _V  ->  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )  =  (/) )
129, 11eqtr4d 2439 . . 3  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
138, 12pm2.61i 158 . 2  |-  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )
14 oduval.d . 2  |-  D  =  (ODual `  O )
15 oduval.l . . . . 5  |-  .<_  =  ( le `  O )
1615cnveqi 5006 . . . 4  |-  `'  .<_  =  `' ( le `  O )
1716opeq2i 3948 . . 3  |-  <. ( le `  ndx ) ,  `'  .<_  >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >.
1817oveq2i 6051 . 2  |-  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >.
)  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)
1913, 14, 183eqtr4i 2434 1  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   <.cop 3777   `'ccnv 4836   ` cfv 5413  (class class class)co 6040   ndxcnx 13421   sSet csts 13422   lecple 13491  ODualcodu 14510
This theorem is referenced by:  oduleval  14513  odubas  14515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-sets 13430  df-odu 14511
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