Definition df-odu 18212

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18216, oduleval 18214, and oduleg 18215 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18420. (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 18211	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3440	. . 3 class V
4	2	cv 1540	. . . 4 class 𝑤
5		cnx 17122	. . . . . 6 class ndx
6		cple 17186	. . . . . 6 class le
7	5, 6	cfv 6492	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6492	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5623	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4586	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17092	. . . 4 class sSet
12	4, 10, 11	co 7358	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5179	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1541	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18213