Definition df-odu 18345

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18349, oduleval 18347, and oduleg 18348 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18553. (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 18344	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3463	. . 3 class V
4	2	cv 1566	. . . 4 class 𝑤
5		cnx 17255	. . . . . 6 class ndx
6		cple 17319	. . . . . 6 class le
7	5, 6	cfv 6539	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6539	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5663	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4600	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17225	. . . 4 class sSet
12	4, 10, 11	co 7413	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5196	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1567	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18346