Definition df-odu 17855

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17859, oduleval 17857, and oduleg 17858 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17914. (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 17854	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3398	. . 3 class V
4	2	cv 1541	. . . 4 class 𝑤
5		cnx 16583	. . . . . 6 class ndx
6		cple 16675	. . . . . 6 class le
7	5, 6	cfv 6339	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6339	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5524	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4522	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 16584	. . . 4 class sSet
12	4, 10, 11	co 7170	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5110	. 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  17856