Definition df-odu 18239

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18243, oduleval 18241, and oduleg 18242 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18447. (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 18238	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3474	. . 3 class V
4	2	cv 1540	. . . 4 class 𝑤
5		cnx 17125	. . . . . 6 class ndx
6		cple 17203	. . . . . 6 class le
7	5, 6	cfv 6543	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6543	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5675	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4634	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17095	. . . 4 class sSet
12	4, 10, 11	co 7408	. . 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 1541	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18240