Definition df-odu 18193

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18197, oduleval 18195, and oduleg 18196 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18401. (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 18192	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3436	. . 3 class V
4	2	cv 1540	. . . 4 class 𝑤
5		cnx 17104	. . . . . 6 class ndx
6		cple 17168	. . . . . 6 class le
7	5, 6	cfv 6481	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6481	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5613	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4579	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17074	. . . 4 class sSet
12	4, 10, 11	co 7346	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5170	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1541	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18194