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Theorem oduval 17062
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 6153 . . . . . . 7 (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
32cnveqd 5263 . . . . . 6 (𝑎 = 𝑂(le‘𝑎) = (le‘𝑂))
43opeq2d 4382 . . . . 5 (𝑎 = 𝑂 → ⟨(le‘ndx), (le‘𝑎)⟩ = ⟨(le‘ndx), (le‘𝑂)⟩)
51, 4oveq12d 6628 . . . 4 (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
6 df-odu 17061 . . . 4 ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩))
7 ovex 6638 . . . 4 (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) ∈ V
85, 6, 7fvmpt 6244 . . 3 (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
9 fvprc 6147 . . . 4 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10 reldmsets 15818 . . . . 5 Rel dom sSet
1110ovprc1 6644 . . . 4 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) = ∅)
129, 11eqtr4d 2658 . . 3 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
138, 12pm2.61i 176 . 2 (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
14 oduval.d . 2 𝐷 = (ODual‘𝑂)
15 oduval.l . . . . 5 = (le‘𝑂)
1615cnveqi 5262 . . . 4 = (le‘𝑂)
1716opeq2i 4379 . . 3 ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), (le‘𝑂)⟩
1817oveq2i 6621 . 2 (𝑂 sSet ⟨(le‘ndx), ⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
1913, 14, 183eqtr4i 2653 1 𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3189  c0 3896  cop 4159  ccnv 5078  cfv 5852  (class class class)co 6610  ndxcnx 15789   sSet csts 15790  lecple 15880  ODualcodu 17060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-sets 15798  df-odu 17061
This theorem is referenced by:  oduleval  17063  odubas  17065
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