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Mirrors > Home > MPE Home > Th. List > df-rngc | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-rngc | ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngc 20510 | . 2 class RngCat | |
2 | vu | . . 3 setvar 𝑢 | |
3 | cvv 3468 | . . 3 class V | |
4 | 2 | cv 1532 | . . . . 5 class 𝑢 |
5 | cestrc 18083 | . . . . 5 class ExtStrCat | |
6 | 4, 5 | cfv 6536 | . . . 4 class (ExtStrCat‘𝑢) |
7 | crnghm 20334 | . . . . 5 class RngHom | |
8 | crng 20055 | . . . . . . 7 class Rng | |
9 | 4, 8 | cin 3942 | . . . . . 6 class (𝑢 ∩ Rng) |
10 | 9, 9 | cxp 5667 | . . . . 5 class ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) |
11 | 7, 10 | cres 5671 | . . . 4 class ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) |
12 | cresc 17762 | . . . 4 class ↾cat | |
13 | 6, 11, 12 | co 7404 | . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) |
14 | 2, 3, 13 | cmpt 5224 | . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) |
15 | 1, 14 | wceq 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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