Definition df-odu 17731

Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 17735, oduleval 17733, and oduleg 17734 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 17790. (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 17730	. 2 class ODual
2		vw	. . 3 setvar 𝑤
3		cvv 3441	. . 3 class V
4	2	cv 1537	. . . 4 class 𝑤
5		cnx 16472	. . . . . 6 class ndx
6		cple 16564	. . . . . 6 class le
7	5, 6	cfv 6324	. . . . 5 class (le‘ndx)
8	4, 6	cfv 6324	. . . . . 6 class (le‘𝑤)
9	8	ccnv 5518	. . . . 5 class ◡(le‘𝑤)
10	7, 9	cop 4531	. . . 4 class ⟨(le‘ndx), ◡(le‘𝑤)⟩
11		csts 16473	. . . 4 class sSet
12	4, 10, 11	co 7135	. . 3 class (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩)
13	2, 3, 12	cmpt 5110	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))
14	1, 13	wceq 1538	1 wff ODual = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(le‘ndx), ◡(le‘𝑤)⟩))

This definition is referenced by:  oduval  17732