Definition df-ringc 45563

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

Step	Hyp	Ref	Expression
1		cringc 45561	. 2 class RingCat
2		vu	. . 3 setvar 𝑢
3		cvv 3432	. . 3 class V
4	2	cv 1538	. . . . 5 class 𝑢
5		cestrc 17838	. . . . 5 class ExtStrCat
6	4, 5	cfv 6433	. . . 4 class (ExtStrCat‘𝑢)
7		crh 19956	. . . . 5 class RingHom
8		crg 19783	. . . . . . 7 class Ring
9	4, 8	cin 3886	. . . . . 6 class (𝑢 ∩ Ring)
10	9, 9	cxp 5587	. . . . 5 class ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))
11	7, 10	cres 5591	. . . 4 class ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))
12		cresc 17520	. . . 4 class ↾cat
13	6, 11, 12	co 7275	. . 3 class ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))
14	2, 3, 13	cmpt 5157	. 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
15	1, 14	wceq 1539	1 wff RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))

This definition is referenced by:  ringcval  45566