Definition df-odu 18357

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18361, oduleval 18359, and oduleg 18360 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18565. (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 18356	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3488	. . 3 class V
4	2	cv 1536	. . . 4 class 𝑤
5		cnx 17240	. . . . . 6 class ndx
6		cple 17318	. . . . . 6 class le
7	5, 6	cfv 6573	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6573	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5699	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4654	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17210	. . . 4 class sSet
12	4, 10, 11	co 7448	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5249	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1537	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18358