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Theorem oduval 17351
 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 6353 . . . . . . 7 (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
32cnveqd 5453 . . . . . 6 (𝑎 = 𝑂(le‘𝑎) = (le‘𝑂))
43opeq2d 4560 . . . . 5 (𝑎 = 𝑂 → ⟨(le‘ndx), (le‘𝑎)⟩ = ⟨(le‘ndx), (le‘𝑂)⟩)
51, 4oveq12d 6832 . . . 4 (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
6 df-odu 17350 . . . 4 ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), (le‘𝑎)⟩))
7 ovex 6842 . . . 4 (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) ∈ V
85, 6, 7fvmpt 6445 . . 3 (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩))
9 fvprc 6347 . . . 4 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10 reldmsets 16108 . . . . 5 Rel dom sSet
1110ovprc1 6848 . . . 4 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩) = ∅)
129, 11eqtr4d 2797 . . 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 5452 . . . 4 = (le‘𝑂)
1716opeq2i 4557 . . 3 ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), (le‘𝑂)⟩
1817oveq2i 6825 . 2 (𝑂 sSet ⟨(le‘ndx), ⟩) = (𝑂 sSet ⟨(le‘ndx), (le‘𝑂)⟩)
1913, 14, 183eqtr4i 2792 1 𝐷 = (𝑂 sSet ⟨(le‘ndx), ⟩)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  ⟨cop 4327  ◡ccnv 5265  ‘cfv 6049  (class class class)co 6814  ndxcnx 16076   sSet csts 16077  lecple 16170  ODualcodu 17349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-sets 16086  df-odu 17350 This theorem is referenced by:  oduleval  17352  odubas  17354
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