Definition df-odu 18344

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18348, oduleval 18346, and oduleg 18347 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18553. (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 18343	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3478	. . 3 class V
4	2	cv 1536	. . . 4 class 𝑤
5		cnx 17227	. . . . . 6 class ndx
6		cple 17305	. . . . . 6 class le
7	5, 6	cfv 6563	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6563	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5688	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4637	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17197	. . . 4 class sSet
12	4, 10, 11	co 7431	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5231	. 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  18345