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Definition df-unit 18408
Description: Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Assertion
Ref Expression
df-unit Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))

Detailed syntax breakdown of Definition df-unit
StepHypRef Expression
1 cui 18405 . 2 class Unit
2 vw . . 3 setvar 𝑤
3 cvv 3169 . . 3 class V
42cv 1473 . . . . . . 7 class 𝑤
5 cdsr 18404 . . . . . . 7 class r
64, 5cfv 5787 . . . . . 6 class (∥r𝑤)
7 coppr 18388 . . . . . . . 8 class oppr
84, 7cfv 5787 . . . . . . 7 class (oppr𝑤)
98, 5cfv 5787 . . . . . 6 class (∥r‘(oppr𝑤))
106, 9cin 3535 . . . . 5 class ((∥r𝑤) ∩ (∥r‘(oppr𝑤)))
1110ccnv 5024 . . . 4 class ((∥r𝑤) ∩ (∥r‘(oppr𝑤)))
12 cur 18267 . . . . . 6 class 1r
134, 12cfv 5787 . . . . 5 class (1r𝑤)
1413csn 4121 . . . 4 class {(1r𝑤)}
1511, 14cima 5028 . . 3 class (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)})
162, 3, 15cmpt 4634 . 2 class (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
171, 16wceq 1474 1 wff Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
Colors of variables: wff setvar class
This definition is referenced by:  isunit  18423
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