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Theorem oduval 16454
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 5879 . . . . . . 7  |-  ( a  =  O  ->  ( le `  a )  =  ( le `  O
) )
32cnveqd 5015 . . . . . 6  |-  ( a  =  O  ->  `' ( le `  a )  =  `' ( le
`  O ) )
43opeq2d 4165 . . . . 5  |-  ( a  =  O  ->  <. ( le `  ndx ) ,  `' ( le `  a ) >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >. )
51, 4oveq12d 6326 . . . 4  |-  ( a  =  O  ->  (
a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
6 df-odu 16453 . . . 4  |- ODual  =  ( a  e.  _V  |->  ( a sSet  <. ( le `  ndx ) ,  `' ( le `  a )
>. ) )
7 ovex 6336 . . . 4  |-  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)  e.  _V
85, 6, 7fvmpt 5963 . . 3  |-  ( O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
9 fvprc 5873 . . . 4  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  (/) )
10 reldmsets 15222 . . . . 5  |-  Rel  dom sSet
1110ovprc1 6339 . . . 4  |-  ( -.  O  e.  _V  ->  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )  =  (/) )
129, 11eqtr4d 2508 . . 3  |-  ( -.  O  e.  _V  ->  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. ) )
138, 12pm2.61i 169 . 2  |-  (ODual `  O )  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O )
>. )
14 oduval.d . 2  |-  D  =  (ODual `  O )
15 oduval.l . . . . 5  |-  .<_  =  ( le `  O )
1615cnveqi 5014 . . . 4  |-  `'  .<_  =  `' ( le `  O )
1716opeq2i 4162 . . 3  |-  <. ( le `  ndx ) ,  `'  .<_  >.  =  <. ( le `  ndx ) ,  `' ( le `  O ) >.
1817oveq2i 6319 . 2  |-  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >.
)  =  ( O sSet  <. ( le `  ndx ) ,  `' ( le `  O ) >.
)
1913, 14, 183eqtr4i 2503 1  |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   <.cop 3965   `'ccnv 4838   ` cfv 5589  (class class class)co 6308   ndxcnx 15196   sSet csts 15197   lecple 15275  ODualcodu 16452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-sets 15205  df-odu 16453
This theorem is referenced by:  oduleval  16455  odubas  16457
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