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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngc 46329 | . 2 class RngCat | |
2 | vu | . . 3 setvar 𝑢 | |
3 | cvv 3448 | . . 3 class V | |
4 | 2 | cv 1541 | . . . . 5 class 𝑢 |
5 | cestrc 18016 | . . . . 5 class ExtStrCat | |
6 | 4, 5 | cfv 6501 | . . . 4 class (ExtStrCat‘𝑢) |
7 | crngh 46257 | . . . . 5 class RngHomo | |
8 | crng 46246 | . . . . . . 7 class Rng | |
9 | 4, 8 | cin 3914 | . . . . . 6 class (𝑢 ∩ Rng) |
10 | 9, 9 | cxp 5636 | . . . . 5 class ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) |
11 | 7, 10 | cres 5640 | . . . 4 class ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) |
12 | cresc 17698 | . . . 4 class ↾cat | |
13 | 6, 11, 12 | co 7362 | . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) |
14 | 2, 3, 13 | cmpt 5193 | . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) |
15 | 1, 14 | wceq 1542 | 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 46334 |
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