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Definition df-setc 17326
Description: Definition of the category Set, relativized to a subset 𝑢. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in 𝑢 and functions between these sets. 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 FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
df-setc SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩})
Distinct variable group:   𝑓,𝑔,𝑢,𝑣,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-setc
StepHypRef Expression
1 csetc 17325 . 2 class SetCat
2 vu . . 3 setvar 𝑢
3 cvv 3495 . . 3 class V
4 cnx 16470 . . . . . 6 class ndx
5 cbs 16473 . . . . . 6 class Base
64, 5cfv 6349 . . . . 5 class (Base‘ndx)
72cv 1527 . . . . 5 class 𝑢
86, 7cop 4565 . . . 4 class ⟨(Base‘ndx), 𝑢
9 chom 16566 . . . . . 6 class Hom
104, 9cfv 6349 . . . . 5 class (Hom ‘ndx)
11 vx . . . . . 6 setvar 𝑥
12 vy . . . . . 6 setvar 𝑦
1312cv 1527 . . . . . . 7 class 𝑦
1411cv 1527 . . . . . . 7 class 𝑥
15 cmap 8396 . . . . . . 7 class m
1613, 14, 15co 7145 . . . . . 6 class (𝑦m 𝑥)
1711, 12, 7, 7, 16cmpo 7147 . . . . 5 class (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))
1810, 17cop 4565 . . . 4 class ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩
19 cco 16567 . . . . . 6 class comp
204, 19cfv 6349 . . . . 5 class (comp‘ndx)
21 vv . . . . . 6 setvar 𝑣
22 vz . . . . . 6 setvar 𝑧
237, 7cxp 5547 . . . . . 6 class (𝑢 × 𝑢)
24 vg . . . . . . 7 setvar 𝑔
25 vf . . . . . . 7 setvar 𝑓
2622cv 1527 . . . . . . . 8 class 𝑧
2721cv 1527 . . . . . . . . 9 class 𝑣
28 c2nd 7679 . . . . . . . . 9 class 2nd
2927, 28cfv 6349 . . . . . . . 8 class (2nd𝑣)
3026, 29, 15co 7145 . . . . . . 7 class (𝑧m (2nd𝑣))
31 c1st 7678 . . . . . . . . 9 class 1st
3227, 31cfv 6349 . . . . . . . 8 class (1st𝑣)
3329, 32, 15co 7145 . . . . . . 7 class ((2nd𝑣) ↑m (1st𝑣))
3424cv 1527 . . . . . . . 8 class 𝑔
3525cv 1527 . . . . . . . 8 class 𝑓
3634, 35ccom 5553 . . . . . . 7 class (𝑔𝑓)
3724, 25, 30, 33, 36cmpo 7147 . . . . . 6 class (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))
3821, 22, 23, 7, 37cmpo 7147 . . . . 5 class (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))
3920, 38cop 4565 . . . 4 class ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩
408, 18, 39ctp 4563 . . 3 class {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩}
412, 3, 40cmpt 5138 . 2 class (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩})
421, 41wceq 1528 1 wff SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  setcval  17327
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