Definition df-odu 18287

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18291, oduleval 18289, and oduleg 18290 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18495. (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 18286	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3461	. . 3 class V
4	2	cv 1532	. . . 4 class 𝑤
5		cnx 17170	. . . . . 6 class ndx
6		cple 17248	. . . . . 6 class le
7	5, 6	cfv 6549	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6549	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5677	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4636	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17140	. . . 4 class sSet
12	4, 10, 11	co 7419	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5232	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1533	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18288