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Theorem oduval 18345
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 = 𝑂𝑎 = 𝑂)
2 fveq2 6907 . . . . . . 7 (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
32cnveqd 5889 . . . . . 6 (𝑎 = 𝑂(le‘𝑎) = (le‘𝑂))
43opeq2d 4885 . . . . 5 (𝑎 = 𝑂 → ⟨(le‘ndx), (le‘𝑎)⟩ = ⟨(le‘ndx), (le‘𝑂)⟩)
51, 4oveq12d 7449 . . . 4 (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
6 df-odu 18344 . . . 4 ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩))
7 ovex 7464 . . . 4 (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) ∈ V
85, 6, 7fvmpt 7016 . . 3 (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
9 fvprc 6899 . . . 4 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10 reldmsets 17199 . . . . 5 Rel dom sSet
1110ovprc1 7470 . . . 4 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) = ∅)
129, 11eqtr4d 2778 . . 3 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
138, 12pm2.61i 182 . 2 (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
14 oduval.d . 2 𝐷 = (ODual‘𝑂)
15 oduval.l . . . . 5 = (le‘𝑂)
1615cnveqi 5888 . . . 4 = (le‘𝑂)
1716opeq2i 4882 . . 3 ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), (le‘𝑂)⟩
1817oveq2i 7442 . 2 (𝑂 sSet ⟨(le‘ndx), ⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
1913, 14, 183eqtr4i 2773 1 𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cop 4637  ccnv 5688  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  lecple 17305  ODualcodu 18343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17198  df-odu 18344
This theorem is referenced by:  oduleval  18346  odubas  18348  odubasOLD  18349
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