Definition df-odu 18222

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18226, oduleval 18224, and oduleg 18225 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18430. (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 18221	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3442	. . 3 class V
4	2	cv 1541	. . . 4 class 𝑤
5		cnx 17132	. . . . . 6 class ndx
6		cple 17196	. . . . . 6 class le
7	5, 6	cfv 6500	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6500	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5631	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4588	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17102	. . . 4 class sSet
12	4, 10, 11	co 7368	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5181	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1542	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18223