Theorem oduval 18366

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 23	. . . . 5 ⊢ (𝑎 = 𝑂 → 𝑎 = 𝑂)
2		fveq2 6885	. . . . . . 7 ⊢ (𝑎 = 𝑂 → (le‘𝑎) = (le‘𝑂))
3	2	cnveqd 5863	. . . . . 6 ⊢ (𝑎 = 𝑂 → ◡(le‘𝑎) = ◡(le‘𝑂))
4	3	opeq2d 4847	. . . . 5 ⊢ (𝑎 = 𝑂 → ⟨(le‘ndx), ◡(le‘𝑎)⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩)
5	1, 4	oveq12d 7437	. . . 4 ⊢ (𝑎 = 𝑂 → (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
6		df-odu 18365	. . . 4 ⊢ ODual = (𝑎 ∈ V ↦ (𝑎 sSet ⟨(le‘ndx), ◡(le‘𝑎)⟩))
7		ovex 7452	. . . 4 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) ∈ V
8	5, 6, 7	fvmpt 6993	. . 3 ⊢ (𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
9		fvprc 6877	. . . 4 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = ∅)
10		reldmsets 17247	. . . . 5 ⊢ Rel dom sSet
11	10	ovprc1 7458	. . . 4 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩) = ∅)
12	9, 11	eqtr4d 2803	. . 3 ⊢ (¬ 𝑂 ∈ V → (ODual‘𝑂) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩))
13	8, 12	pm2.61i 184	. 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 5862	. . . 4 ⊢ ◡ ≤ = ◡(le‘𝑂)
17	16	opeq2i 4844	. . 3 ⊢ ⟨(le‘ndx), ◡ ≤ ⟩ = ⟨(le‘ndx), ◡(le‘𝑂)⟩
18	17	oveq2i 7430	. 2 ⊢ (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩) = (𝑂 sSet ⟨(le‘ndx), ◡(le‘𝑂)⟩)
19	13, 14, 18	3eqtr4i 2798	1 ⊢ 𝐷 = (𝑂 sSet ⟨(le‘ndx), ◡ ≤ ⟩)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  ⟨cop 4597  ^◡ccnv 5662  ‘cfv 6540  (class class class)co 7419   sSet csts 17245  ndxcnx 17275  lecple 17339  ODualcodu 18364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-sets 17246  df-odu 18365

This theorem is used by:  oduleval  18367  odubas  18369