Definition df-rngc 45405

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

Step	Hyp	Ref	Expression
1		crngc 45403	. 2 class RngCat
2		vu	. . 3 setvar 𝑢
3		cvv 3422	. . 3 class V
4	2	cv 1538	. . . . 5 class 𝑢
5		cestrc 17754	. . . . 5 class ExtStrCat
6	4, 5	cfv 6418	. . . 4 class (ExtStrCat‘𝑢)
7		crngh 45331	. . . . 5 class RngHomo
8		crng 45320	. . . . . . 7 class Rng
9	4, 8	cin 3882	. . . . . 6 class (𝑢 ∩ Rng)
10	9, 9	cxp 5578	. . . . 5 class ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))
11	7, 10	cres 5582	. . . 4 class ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))
12		cresc 17437	. . . 4 class ↾cat
13	6, 11, 12	co 7255	. . 3 class ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))
14	2, 3, 13	cmpt 5153	. 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
15	1, 14	wceq 1539	1 wff RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))

This definition is referenced by:  rngcval  45408