Definition df-odu 18377

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18381, oduleval 18379, and oduleg 18380 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18585. (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 18376	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3453	. . 3 class V
4	2	cv 1569	. . . 4 class 𝑤
5		cnx 17287	. . . . . 6 class ndx
6		cple 17351	. . . . . 6 class le
7	5, 6	cfv 6537	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6537	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5658	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4593	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17257	. . . 4 class sSet
12	4, 10, 11	co 7416	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5190	. 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  18378