Metamath Proof Explorer

Definition df-ringc

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
Assertion df-ringc ⊢ RingCat = u ∈ V ⟼ ExtStrCat ⁡ u ↾ cat RingHom ↾ u ∩ Ring × u ∩ Ring

Detailed syntax breakdown

Step Hyp Ref Expression
0 cringc class RingCat
1 vu setvar u
2 cvv class V
3 cestrc class ExtStrCat
4 1 cv setvar u
5 4 3 cfv class ExtStrCat ⁡ u
6 cresc class ↾ cat
7 crh class RingHom
8 crg class Ring
9 4 8 cin class u ∩ Ring
10 9 9 cxp class u ∩ Ring × u ∩ Ring
11 7 10 cres class RingHom ↾ u ∩ Ring × u ∩ Ring
12 5 11 6 co class ExtStrCat ⁡ u ↾ cat RingHom ↾ u ∩ Ring × u ∩ Ring
13 1 2 12 cmpt class u ∈ V ⟼ ExtStrCat ⁡ u ↾ cat RingHom ↾ u ∩ Ring × u ∩ Ring
14 0 13 wceq wff RingCat = u ∈ V ⟼ ExtStrCat ⁡ u ↾ cat RingHom ↾ u ∩ Ring × u ∩ Ring