Definition df-odu 17444

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17448, oduleval 17446, and oduleg 17447 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17503. (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 17443	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3385	. . 3 class V
4	2	cv 1652	. . . 4 class 𝑤
5		cnx 16181	. . . . . 6 class ndx
6		cple 16274	. . . . . 6 class le
7	5, 6	cfv 6101	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6101	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5311	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4374	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 16182	. . . 4 class sSet
12	4, 10, 11	co 6878	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 4922	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1653	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  17445