Definition df-odu 18245

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18249, oduleval 18247, and oduleg 18248 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18453. (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 18244	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3431	. . 3 class V
4	2	cv 1546	. . . 4 class 𝑤
5		cnx 17155	. . . . . 6 class ndx
6		cple 17219	. . . . . 6 class le
7	5, 6	cfv 6486	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6486	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5618	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4562	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17125	. . . 4 class sSet
12	4, 10, 11	co 7357	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5154	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1547	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18246