Definition df-odu 17986

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17990, oduleval 17988, and oduleg 17989 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18194. (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 17985	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3430	. . 3 class V
4	2	cv 1540	. . . 4 class 𝑤
5		cnx 16875	. . . . . 6 class ndx
6		cple 16950	. . . . . 6 class le
7	5, 6	cfv 6430	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6430	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5587	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4572	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 16845	. . . 4 class sSet
12	4, 10, 11	co 7268	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5161	. 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  17987