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 1539	. . . 4 class 𝑤
5		cnx 17104	. . . . . 6 class ndx
6		cple 17168	. . . . . 6 class le
7	5, 6	cfv 6482	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6482	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5618	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4583	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17074	. . . 4 class sSet
12	4, 10, 11	co 7349	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5173	. 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  18194