Definition df-odu 18253

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 18257, oduleval 18255, and oduleg 18256 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 18461. (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 18252	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3429	. . 3 class V
4	2	cv 1541	. . . 4 class 𝑤
5		cnx 17163	. . . . . 6 class ndx
6		cple 17227	. . . . . 6 class le
7	5, 6	cfv 6498	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6498	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5630	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4573	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 17133	. . . 4 class sSet
12	4, 10, 11	co 7367	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5166	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1542	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  18254