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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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