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Definition df-catc 18048
Description: Definition of the category Cat, which consists of all categories in the universe 𝑒 (i.e., "𝑒-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 "𝑒 -large categories", which can be expressed as (Cat βˆ– 𝑒). (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
df-catc CatCat = (𝑒 ∈ V ↦ ⦋(𝑒 ∩ Cat) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩})
Distinct variable group:   𝑓,𝑏,𝑔,𝑒,𝑣,π‘₯,𝑦,𝑧

Detailed syntax breakdown of Definition df-catc
StepHypRef Expression
1 ccatc 18047 . 2 class CatCat
2 vu . . 3 setvar 𝑒
3 cvv 3474 . . 3 class V
4 vb . . . 4 setvar 𝑏
52cv 1540 . . . . 5 class 𝑒
6 ccat 17607 . . . . 5 class Cat
75, 6cin 3947 . . . 4 class (𝑒 ∩ Cat)
8 cnx 17125 . . . . . . 7 class ndx
9 cbs 17143 . . . . . . 7 class Base
108, 9cfv 6543 . . . . . 6 class (Baseβ€˜ndx)
114cv 1540 . . . . . 6 class 𝑏
1210, 11cop 4634 . . . . 5 class ⟨(Baseβ€˜ndx), π‘βŸ©
13 chom 17207 . . . . . . 7 class Hom
148, 13cfv 6543 . . . . . 6 class (Hom β€˜ndx)
15 vx . . . . . . 7 setvar π‘₯
16 vy . . . . . . 7 setvar 𝑦
1715cv 1540 . . . . . . . 8 class π‘₯
1816cv 1540 . . . . . . . 8 class 𝑦
19 cfunc 17803 . . . . . . . 8 class Func
2017, 18, 19co 7408 . . . . . . 7 class (π‘₯ Func 𝑦)
2115, 16, 11, 11, 20cmpo 7410 . . . . . 6 class (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))
2214, 21cop 4634 . . . . 5 class ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩
23 cco 17208 . . . . . . 7 class comp
248, 23cfv 6543 . . . . . 6 class (compβ€˜ndx)
25 vv . . . . . . 7 setvar 𝑣
26 vz . . . . . . 7 setvar 𝑧
2711, 11cxp 5674 . . . . . . 7 class (𝑏 Γ— 𝑏)
28 vg . . . . . . . 8 setvar 𝑔
29 vf . . . . . . . 8 setvar 𝑓
3025cv 1540 . . . . . . . . . 10 class 𝑣
31 c2nd 7973 . . . . . . . . . 10 class 2nd
3230, 31cfv 6543 . . . . . . . . 9 class (2nd β€˜π‘£)
3326cv 1540 . . . . . . . . 9 class 𝑧
3432, 33, 19co 7408 . . . . . . . 8 class ((2nd β€˜π‘£) Func 𝑧)
3530, 19cfv 6543 . . . . . . . 8 class ( Func β€˜π‘£)
3628cv 1540 . . . . . . . . 9 class 𝑔
3729cv 1540 . . . . . . . . 9 class 𝑓
38 ccofu 17805 . . . . . . . . 9 class ∘func
3936, 37, 38co 7408 . . . . . . . 8 class (𝑔 ∘func 𝑓)
4028, 29, 34, 35, 39cmpo 7410 . . . . . . 7 class (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓))
4125, 26, 27, 11, 40cmpo 7410 . . . . . 6 class (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))
4224, 41cop 4634 . . . . 5 class ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩
4312, 22, 42ctp 4632 . . . 4 class {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩}
444, 7, 43csb 3893 . . 3 class ⦋(𝑒 ∩ Cat) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩}
452, 3, 44cmpt 5231 . 2 class (𝑒 ∈ V ↦ ⦋(𝑒 ∩ Cat) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩})
461, 45wceq 1541 1 wff CatCat = (𝑒 ∈ V ↦ ⦋(𝑒 ∩ Cat) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ Func 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) Func 𝑧), 𝑓 ∈ ( Func β€˜π‘£) ↦ (𝑔 ∘func 𝑓)))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  catcval  18049
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