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Definition df-catc 16952
 Description: Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e. "𝑢-small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "𝑢 -large categories", which can be expressed as (Cat ∖ 𝑢). (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
df-catc CatCat = (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
Distinct variable group:   𝑓,𝑏,𝑔,𝑢,𝑣,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-catc
StepHypRef Expression
1 ccatc 16951 . 2 class CatCat
2 vu . . 3 setvar 𝑢
3 cvv 3351 . . 3 class V
4 vb . . . 4 setvar 𝑏
52cv 1630 . . . . 5 class 𝑢
6 ccat 16532 . . . . 5 class Cat
75, 6cin 3722 . . . 4 class (𝑢 ∩ Cat)
8 cnx 16061 . . . . . . 7 class ndx
9 cbs 16064 . . . . . . 7 class Base
108, 9cfv 6031 . . . . . 6 class (Base‘ndx)
114cv 1630 . . . . . 6 class 𝑏
1210, 11cop 4322 . . . . 5 class ⟨(Base‘ndx), 𝑏
13 chom 16160 . . . . . . 7 class Hom
148, 13cfv 6031 . . . . . 6 class (Hom ‘ndx)
15 vx . . . . . . 7 setvar 𝑥
16 vy . . . . . . 7 setvar 𝑦
1715cv 1630 . . . . . . . 8 class 𝑥
1816cv 1630 . . . . . . . 8 class 𝑦
19 cfunc 16721 . . . . . . . 8 class Func
2017, 18, 19co 6793 . . . . . . 7 class (𝑥 Func 𝑦)
2115, 16, 11, 11, 20cmpt2 6795 . . . . . 6 class (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))
2214, 21cop 4322 . . . . 5 class ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩
23 cco 16161 . . . . . . 7 class comp
248, 23cfv 6031 . . . . . 6 class (comp‘ndx)
25 vv . . . . . . 7 setvar 𝑣
26 vz . . . . . . 7 setvar 𝑧
2711, 11cxp 5247 . . . . . . 7 class (𝑏 × 𝑏)
28 vg . . . . . . . 8 setvar 𝑔
29 vf . . . . . . . 8 setvar 𝑓
3025cv 1630 . . . . . . . . . 10 class 𝑣
31 c2nd 7314 . . . . . . . . . 10 class 2nd
3230, 31cfv 6031 . . . . . . . . 9 class (2nd𝑣)
3326cv 1630 . . . . . . . . 9 class 𝑧
3432, 33, 19co 6793 . . . . . . . 8 class ((2nd𝑣) Func 𝑧)
3530, 19cfv 6031 . . . . . . . 8 class ( Func ‘𝑣)
3628cv 1630 . . . . . . . . 9 class 𝑔
3729cv 1630 . . . . . . . . 9 class 𝑓
38 ccofu 16723 . . . . . . . . 9 class func
3936, 37, 38co 6793 . . . . . . . 8 class (𝑔func 𝑓)
4028, 29, 34, 35, 39cmpt2 6795 . . . . . . 7 class (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))
4125, 26, 27, 11, 40cmpt2 6795 . . . . . 6 class (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))
4224, 41cop 4322 . . . . 5 class ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩
4312, 22, 42ctp 4320 . . . 4 class {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩}
444, 7, 43csb 3682 . . 3 class (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩}
452, 3, 44cmpt 4863 . 2 class (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
461, 45wceq 1631 1 wff CatCat = (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})
 Colors of variables: wff setvar class This definition is referenced by:  catcval  16953
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