Definition df-odu 18228

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18232, oduleval 18230, and oduleg 18231 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18436. (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 18227	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3444	. . 3 class V
4	2	cv 1539	. . . 4 class 𝑤
5		cnx 17139	. . . . . 6 class ndx
6		cple 17203	. . . . . 6 class le
7	5, 6	cfv 6499	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6499	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5630	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4591	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17109	. . . 4 class sSet
12	4, 10, 11	co 7369	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5183	. 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  18229