Definition df-odu 18248

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18252, oduleval 18250, and oduleg 18251 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18456. (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 18247	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3430	. . 3 class V
4	2	cv 1541	. . . 4 class 𝑤
5		cnx 17158	. . . . . 6 class ndx
6		cple 17222	. . . . . 6 class le
7	5, 6	cfv 6494	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6494	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5625	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4574	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17128	. . . 4 class sSet
12	4, 10, 11	co 7362	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5167	. 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  18249