Definition df-odu 18461

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18465, oduleval 18463, and oduleg 18464 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18669. (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 18460	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3450	. . 3 class V
4	2	cv 1568	. . . 4 class 𝑤
5		cnx 17371	. . . . . 6 class ndx
6		cple 17435	. . . . . 6 class le
7	5, 6	cfv 6538	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6538	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5649	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4589	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17341	. . . 4 class sSet
12	4, 10, 11	co 7420	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5185	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1569	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is used by:  oduval  18462