Definition df-odu 18347

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18351, oduleval 18349, and oduleg 18350 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18555. (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 18346	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3455	. . 3 class V
4	2	cv 1569	. . . 4 class 𝑤
5		cnx 17257	. . . . . 6 class ndx
6		cple 17321	. . . . . 6 class le
7	5, 6	cfv 6536	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6536	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5660	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4595	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17227	. . . 4 class sSet
12	4, 10, 11	co 7410	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5192	. 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  18348