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Definition df-rngc 41749
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 ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))

Detailed syntax breakdown of Definition df-rngc
StepHypRef Expression
1 crngc 41747 . 2 class RngCat
2 vu . . 3 setvar 𝑢
3 cvv 3168 . . 3 class V
42cv 1473 . . . . 5 class 𝑢
5 cestrc 16527 . . . . 5 class ExtStrCat
64, 5cfv 5786 . . . 4 class (ExtStrCat‘𝑢)
7 crngh 41673 . . . . 5 class RngHomo
8 crng 41662 . . . . . . 7 class Rng
94, 8cin 3534 . . . . . 6 class (𝑢 ∩ Rng)
109, 9cxp 5022 . . . . 5 class ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))
117, 10cres 5026 . . . 4 class ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))
12 cresc 16233 . . . 4 class cat
136, 11, 12co 6523 . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))
142, 3, 13cmpt 4633 . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
151, 14wceq 1474 1 wff RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
Colors of variables: wff setvar class
This definition is referenced by:  rngcval  41752
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