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Definition df-setc 16495
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 16494 . 2 class SetCat
2 vu . . 3 setvar 𝑢
3 cvv 3172 . . 3 class V
4 cnx 15638 . . . . . 6 class ndx
5 cbs 15641 . . . . . 6 class Base
64, 5cfv 5790 . . . . 5 class (Base‘ndx)
72cv 1473 . . . . 5 class 𝑢
86, 7cop 4130 . . . 4 class ⟨(Base‘ndx), 𝑢
9 chom 15725 . . . . . 6 class Hom
104, 9cfv 5790 . . . . 5 class (Hom ‘ndx)
11 vx . . . . . 6 setvar 𝑥
12 vy . . . . . 6 setvar 𝑦
1312cv 1473 . . . . . . 7 class 𝑦
1411cv 1473 . . . . . . 7 class 𝑥
15 cmap 7721 . . . . . . 7 class 𝑚
1613, 14, 15co 6527 . . . . . 6 class (𝑦𝑚 𝑥)
1711, 12, 7, 7, 16cmpt2 6529 . . . . 5 class (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))
1810, 17cop 4130 . . . 4 class ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))⟩
19 cco 15726 . . . . . 6 class comp
204, 19cfv 5790 . . . . 5 class (comp‘ndx)
21 vv . . . . . 6 setvar 𝑣
22 vz . . . . . 6 setvar 𝑧
237, 7cxp 5026 . . . . . 6 class (𝑢 × 𝑢)
24 vg . . . . . . 7 setvar 𝑔
25 vf . . . . . . 7 setvar 𝑓
2622cv 1473 . . . . . . . 8 class 𝑧
2721cv 1473 . . . . . . . . 9 class 𝑣
28 c2nd 7035 . . . . . . . . 9 class 2nd
2927, 28cfv 5790 . . . . . . . 8 class (2nd𝑣)
3026, 29, 15co 6527 . . . . . . 7 class (𝑧𝑚 (2nd𝑣))
31 c1st 7034 . . . . . . . . 9 class 1st
3227, 31cfv 5790 . . . . . . . 8 class (1st𝑣)
3329, 32, 15co 6527 . . . . . . 7 class ((2nd𝑣) ↑𝑚 (1st𝑣))
3424cv 1473 . . . . . . . 8 class 𝑔
3525cv 1473 . . . . . . . 8 class 𝑓
3634, 35ccom 5032 . . . . . . 7 class (𝑔𝑓)
3724, 25, 30, 33, 36cmpt2 6529 . . . . . 6 class (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))
3821, 22, 23, 7, 37cmpt2 6529 . . . . 5 class (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))
3920, 38cop 4130 . . . 4 class ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩
408, 18, 39ctp 4128 . . 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 4637 . 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 1474 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  16496
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