Definition df-odu 18254

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18258, oduleval 18256, and oduleg 18257 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18460. (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 18253	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3450	. . 3 class V
4	2	cv 1539	. . . 4 class 𝑤
5		cnx 17169	. . . . . 6 class ndx
6		cple 17233	. . . . . 6 class le
7	5, 6	cfv 6513	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6513	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5639	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4597	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17139	. . . 4 class sSet
12	4, 10, 11	co 7389	. . 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 1540	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18255