Definition df-odu 18240

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18244, oduleval 18242, and oduleg 18243 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18448. (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 18239	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3475	. . 3 class V
4	2	cv 1541	. . . 4 class 𝑤
5		cnx 17126	. . . . . 6 class ndx
6		cple 17204	. . . . . 6 class le
7	5, 6	cfv 6544	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6544	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5676	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4635	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17096	. . . 4 class sSet
12	4, 10, 11	co 7409	. . 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 1542	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18241