Definition df-odu 18304

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18308, oduleval 18306, and oduleg 18307 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18510. (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 18303	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3464	. . 3 class V
4	2	cv 1539	. . . 4 class 𝑤
5		cnx 17217	. . . . . 6 class ndx
6		cple 17283	. . . . . 6 class le
7	5, 6	cfv 6536	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6536	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5658	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4612	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17187	. . . 4 class sSet
12	4, 10, 11	co 7410	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5206	. 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  18305