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Definition df-rngc 20511
Description: Definition of the category Rng, relativized to a subset 𝑢. This is the category of all non-unital rings in 𝑢 and homomorphisms between these rings. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Assertion
Ref Expression
df-rngc RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))

Detailed syntax breakdown of Definition df-rngc
StepHypRef Expression
1 crngc 20510 . 2 class RngCat
2 vu . . 3 setvar 𝑢
3 cvv 3468 . . 3 class V
42cv 1532 . . . . 5 class 𝑢
5 cestrc 18083 . . . . 5 class ExtStrCat
64, 5cfv 6536 . . . 4 class (ExtStrCat‘𝑢)
7 crnghm 20334 . . . . 5 class RngHom
8 crng 20055 . . . . . . 7 class Rng
94, 8cin 3942 . . . . . 6 class (𝑢 ∩ Rng)
109, 9cxp 5667 . . . . 5 class ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))
117, 10cres 5671 . . . 4 class ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))
12 cresc 17762 . . . 4 class cat
136, 11, 12co 7404 . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))
142, 3, 13cmpt 5224 . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
151, 14wceq 1533 1 wff RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
Colors of variables: wff setvar class
This definition is referenced by:  rngcval  20512
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