Definition df-odu 18487

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18491, oduleval 18489, and oduleg 18490 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18695. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Assertion

Ref	Expression
df-odu	⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

Detailed syntax breakdown of Definition df-odu

Step	Hyp	Ref	Expression
1		codu 18486	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3454	. . 3 class V
4	2	cv 1569	. . . 4 class 𝑤
5		cnx 17397	. . . . . 6 class ndx
6		cple 17461	. . . . . 6 class le
7	5, 6	cfv 6555	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6555	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5654	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4838	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17367	. . . 4 class sSet
12	4, 10, 11	co 7440	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5189	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1570	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is used by:  oduval  18488