Definition df-odu 17129

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17133, oduleval 17131, and oduleg 17132 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17188. (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 17128	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3200	. . 3 class V
4	2	cv 1482	. . . 4 class 𝑤
5		cnx 15854	. . . . . 6 class ndx
6		cple 15948	. . . . . 6 class le
7	5, 6	cfv 5888	. . . . 5 class (le‘ndx)
8	4, 6	cfv 5888	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5113	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4183	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 15855	. . . 4 class sSet
12	4, 10, 11	co 6650	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 4729	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1483	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  17130