Metamath Proof Explorer


Definition df-setc

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)

Ref Expression
Assertion 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 ) ) ) ⟩ } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 csetc SetCat
1 vu 𝑢
2 cvv V
3 cbs Base
4 cnx ndx
5 4 3 cfv ( Base ‘ ndx )
6 1 cv 𝑢
7 5 6 cop ⟨ ( Base ‘ ndx ) , 𝑢
8 chom Hom
9 4 8 cfv ( Hom ‘ ndx )
10 vx 𝑥
11 vy 𝑦
12 11 cv 𝑦
13 cmap m
14 10 cv 𝑥
15 12 14 13 co ( 𝑦m 𝑥 )
16 10 11 6 6 15 cmpo ( 𝑥𝑢 , 𝑦𝑢 ↦ ( 𝑦m 𝑥 ) )
17 9 16 cop ⟨ ( Hom ‘ ndx ) , ( 𝑥𝑢 , 𝑦𝑢 ↦ ( 𝑦m 𝑥 ) ) ⟩
18 cco comp
19 4 18 cfv ( comp ‘ ndx )
20 vv 𝑣
21 6 6 cxp ( 𝑢 × 𝑢 )
22 vz 𝑧
23 vg 𝑔
24 22 cv 𝑧
25 c2nd 2nd
26 20 cv 𝑣
27 26 25 cfv ( 2nd𝑣 )
28 24 27 13 co ( 𝑧m ( 2nd𝑣 ) )
29 vf 𝑓
30 c1st 1st
31 26 30 cfv ( 1st𝑣 )
32 27 31 13 co ( ( 2nd𝑣 ) ↑m ( 1st𝑣 ) )
33 23 cv 𝑔
34 29 cv 𝑓
35 33 34 ccom ( 𝑔𝑓 )
36 23 29 28 32 35 cmpo ( 𝑔 ∈ ( 𝑧m ( 2nd𝑣 ) ) , 𝑓 ∈ ( ( 2nd𝑣 ) ↑m ( 1st𝑣 ) ) ↦ ( 𝑔𝑓 ) )
37 20 22 21 6 36 cmpo ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧𝑢 ↦ ( 𝑔 ∈ ( 𝑧m ( 2nd𝑣 ) ) , 𝑓 ∈ ( ( 2nd𝑣 ) ↑m ( 1st𝑣 ) ) ↦ ( 𝑔𝑓 ) ) )
38 19 37 cop ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧𝑢 ↦ ( 𝑔 ∈ ( 𝑧m ( 2nd𝑣 ) ) , 𝑓 ∈ ( ( 2nd𝑣 ) ↑m ( 1st𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩
39 7 17 38 ctp { ⟨ ( Base ‘ ndx ) , 𝑢 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥𝑢 , 𝑦𝑢 ↦ ( 𝑦m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧𝑢 ↦ ( 𝑔 ∈ ( 𝑧m ( 2nd𝑣 ) ) , 𝑓 ∈ ( ( 2nd𝑣 ) ↑m ( 1st𝑣 ) ) ↦ ( 𝑔𝑓 ) ) ) ⟩ }
40 1 2 39 cmpt ( 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 ) ) ) ⟩ } )
41 0 40 wceq 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 ) ) ) ⟩ } )