Definition df-odu 18255

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18259, oduleval 18257, and oduleg 18258 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18461. (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 18254	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3450	. . 3 class V
4	2	cv 1539	. . . 4 class 𝑤
5		cnx 17170	. . . . . 6 class ndx
6		cple 17234	. . . . . 6 class le
7	5, 6	cfv 6514	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6514	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5640	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4598	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17140	. . . 4 class sSet
12	4, 10, 11	co 7390	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5191	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1540	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18256