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Definition df-ringc 42533
Description: Definition of the category Ring, relativized to a subset 𝑢. See also the note in [Lang] p. 91, and the item Rng in [Adamek] p. 478. This is the category of all 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, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Assertion
Ref Expression
df-ringc RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))

Detailed syntax breakdown of Definition df-ringc
StepHypRef Expression
1 cringc 42531 . 2 class RingCat
2 vu . . 3 setvar 𝑢
3 cvv 3351 . . 3 class V
42cv 1630 . . . . 5 class 𝑢
5 cestrc 16969 . . . . 5 class ExtStrCat
64, 5cfv 6031 . . . 4 class (ExtStrCat‘𝑢)
7 crh 18922 . . . . 5 class RingHom
8 crg 18755 . . . . . . 7 class Ring
94, 8cin 3722 . . . . . 6 class (𝑢 ∩ Ring)
109, 9cxp 5247 . . . . 5 class ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))
117, 10cres 5251 . . . 4 class ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))
12 cresc 16675 . . . 4 class cat
136, 11, 12co 6793 . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))
142, 3, 13cmpt 4863 . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
151, 14wceq 1631 1 wff RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
Colors of variables: wff setvar class
This definition is referenced by:  ringcval  42536
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