Definition df-odu 18400

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18404, oduleval 18402, and oduleg 18403 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18608. (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 18399	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3450	. . 3 class V
4	2	cv 1569	. . . 4 class 𝑤
5		cnx 17310	. . . . . 6 class ndx
6		cple 17374	. . . . . 6 class le
7	5, 6	cfv 6535	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6535	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5654	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4590	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17280	. . . 4 class sSet
12	4, 10, 11	co 7416	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5186	. 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  18401