Theorem oduval 18333

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.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑎 = 𝑂 → 𝑎 = 𝑂)
2		fveq2 6906	. . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
3	2	cnveqd 5886	. . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂))
4	3	opeq2d 4880	. . . . 5 ⊢ (𝑎 = 𝑂 → ⟨(le‘ndx), ◡(le‘𝑎)⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩)
5	1, 4	oveq12d 7449	. . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
6		df-odu 18332	. . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩))
7		ovex 7464	. . . 4 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) ∈ V
8	5, 6, 7	fvmpt 7016	. . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
9		fvprc 6898	. . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10		reldmsets 17202	. . . . 5 ⊢ Rel dom sSet
11	10	ovprc1 7470	. . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) = ∅)
12	9, 11	eqtr4d 2780	. . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
13	8, 12	pm2.61i 182	. 2 ⊢ (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
14		oduval.d	. 2 ⊢ 𝐷 = (ODual‘𝑂)
15		oduval.l	. . . . 5 ⊢ ≤ = (le‘𝑂)
16	15	cnveqi 5885	. . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂)
17	16	opeq2i 4877	. . 3 ⊢ ⟨(le‘ndx), ◡ ≤ ⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩
18	17	oveq2i 7442	. 2 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
19	13, 14, 18	3eqtr4i 2775	1 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ⟨cop 4632  ^◡ccnv 5684  ‘cfv 6561  (class class class)co 7431   sSet csts 17200  ndxcnx 17230  lecple 17304  ODualcodu 18331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201  df-odu 18332

This theorem is referenced by:  oduleval  18334  odubas  18336  odubasOLD  18337