Definition df-odu 18333

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18337, oduleval 18335, and oduleg 18336 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18541. (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 18332	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3479	. . 3 class V
4	2	cv 1538	. . . 4 class 𝑤
5		cnx 17231	. . . . . 6 class ndx
6		cple 17305	. . . . . 6 class le
7	5, 6	cfv 6560	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6560	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5683	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4631	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17201	. . . 4 class sSet
12	4, 10, 11	co 7432	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5224	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1539	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18334