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Definition df-setc 17967
Description: Definition of the category Set, relativized to a subset 𝑒. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in 𝑒 and functions between these sets. Generally, we will take 𝑒 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 = (𝑒 ∈ V ↦ {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩})
Distinct variable group:   𝑓,𝑔,𝑒,𝑣,π‘₯,𝑦,𝑧

Detailed syntax breakdown of Definition df-setc
StepHypRef Expression
1 csetc 17966 . 2 class SetCat
2 vu . . 3 setvar 𝑒
3 cvv 3444 . . 3 class V
4 cnx 17070 . . . . . 6 class ndx
5 cbs 17088 . . . . . 6 class Base
64, 5cfv 6497 . . . . 5 class (Baseβ€˜ndx)
72cv 1541 . . . . 5 class 𝑒
86, 7cop 4593 . . . 4 class ⟨(Baseβ€˜ndx), π‘’βŸ©
9 chom 17149 . . . . . 6 class Hom
104, 9cfv 6497 . . . . 5 class (Hom β€˜ndx)
11 vx . . . . . 6 setvar π‘₯
12 vy . . . . . 6 setvar 𝑦
1312cv 1541 . . . . . . 7 class 𝑦
1411cv 1541 . . . . . . 7 class π‘₯
15 cmap 8768 . . . . . . 7 class ↑m
1613, 14, 15co 7358 . . . . . 6 class (𝑦 ↑m π‘₯)
1711, 12, 7, 7, 16cmpo 7360 . . . . 5 class (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))
1810, 17cop 4593 . . . 4 class ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))⟩
19 cco 17150 . . . . . 6 class comp
204, 19cfv 6497 . . . . 5 class (compβ€˜ndx)
21 vv . . . . . 6 setvar 𝑣
22 vz . . . . . 6 setvar 𝑧
237, 7cxp 5632 . . . . . 6 class (𝑒 Γ— 𝑒)
24 vg . . . . . . 7 setvar 𝑔
25 vf . . . . . . 7 setvar 𝑓
2622cv 1541 . . . . . . . 8 class 𝑧
2721cv 1541 . . . . . . . . 9 class 𝑣
28 c2nd 7921 . . . . . . . . 9 class 2nd
2927, 28cfv 6497 . . . . . . . 8 class (2nd β€˜π‘£)
3026, 29, 15co 7358 . . . . . . 7 class (𝑧 ↑m (2nd β€˜π‘£))
31 c1st 7920 . . . . . . . . 9 class 1st
3227, 31cfv 6497 . . . . . . . 8 class (1st β€˜π‘£)
3329, 32, 15co 7358 . . . . . . 7 class ((2nd β€˜π‘£) ↑m (1st β€˜π‘£))
3424cv 1541 . . . . . . . 8 class 𝑔
3525cv 1541 . . . . . . . 8 class 𝑓
3634, 35ccom 5638 . . . . . . 7 class (𝑔 ∘ 𝑓)
3724, 25, 30, 33, 36cmpo 7360 . . . . . 6 class (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))
3821, 22, 23, 7, 37cmpo 7360 . . . . 5 class (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))
3920, 38cop 4593 . . . 4 class ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩
408, 18, 39ctp 4591 . . 3 class {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩}
412, 3, 40cmpt 5189 . 2 class (𝑒 ∈ V ↦ {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩})
421, 41wceq 1542 1 wff SetCat = (𝑒 ∈ V ↦ {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ (𝑦 ↑m π‘₯))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ (𝑧 ↑m (2nd β€˜π‘£)), 𝑓 ∈ ((2nd β€˜π‘£) ↑m (1st β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  setcval  17968
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