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Theorem oduval 16953
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 6103 . . . . . . 7 (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
32cnveqd 5220 . . . . . 6 (𝑎 = 𝑂(le‘𝑎) = (le‘𝑂))
43opeq2d 4347 . . . . 5 (𝑎 = 𝑂 → ⟨(le‘ndx), (le‘𝑎)⟩ = ⟨(le‘ndx), (le‘𝑂)⟩)
51, 4oveq12d 6567 . . . 4 (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
6 df-odu 16952 . . . 4 ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩))
7 ovex 6577 . . . 4 (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) ∈ V
85, 6, 7fvmpt 6191 . . 3 (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
9 fvprc 6097 . . . 4 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10 reldmsets 15718 . . . . 5 Rel dom sSet
1110ovprc1 6582 . . . 4 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) = ∅)
129, 11eqtr4d 2647 . . 3 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
138, 12pm2.61i 175 . 2 (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
14 oduval.d . 2 𝐷 = (ODual‘𝑂)
15 oduval.l . . . . 5 = (le‘𝑂)
1615cnveqi 5219 . . . 4 = (le‘𝑂)
1716opeq2i 4344 . . 3 ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), (le‘𝑂)⟩
1817oveq2i 6560 . 2 (𝑂 sSet ⟨(le‘ndx), ⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
1913, 14, 183eqtr4i 2642 1 𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  cop 4131  ccnv 5037  cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  lecple 15775  ODualcodu 16951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-sets 15701  df-odu 16952
This theorem is referenced by:  oduleval  16954  odubas  16956
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