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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-ringc | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-ringc | ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cringc 45530 | . 2 class RingCat | |
2 | vu | . . 3 setvar 𝑢 | |
3 | cvv 3431 | . . 3 class V | |
4 | 2 | cv 1541 | . . . . 5 class 𝑢 |
5 | cestrc 17836 | . . . . 5 class ExtStrCat | |
6 | 4, 5 | cfv 6432 | . . . 4 class (ExtStrCat‘𝑢) |
7 | crh 19954 | . . . . 5 class RingHom | |
8 | crg 19781 | . . . . . . 7 class Ring | |
9 | 4, 8 | cin 3891 | . . . . . 6 class (𝑢 ∩ Ring) |
10 | 9, 9 | cxp 5588 | . . . . 5 class ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) |
11 | 7, 10 | cres 5592 | . . . 4 class ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) |
12 | cresc 17518 | . . . 4 class ↾cat | |
13 | 6, 11, 12 | co 7271 | . . 3 class ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) |
14 | 2, 3, 13 | cmpt 5162 | . 2 class (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))) |
15 | 1, 14 | wceq 1542 | 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 45535 |
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