Definition df-odu 18054

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18058, oduleval 18056, and oduleg 18057 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18262. (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 18053	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3437	. . 3 class V
4	2	cv 1538	. . . 4 class 𝑤
5		cnx 16943	. . . . . 6 class ndx
6		cple 17018	. . . . . 6 class le
7	5, 6	cfv 6458	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6458	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5599	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4571	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 16913	. . . 4 class sSet
12	4, 10, 11	co 7307	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5164	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1539	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18055