MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-estrc Structured version   Visualization version   GIF version

Definition df-estrc 17835
Description: Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe 𝑢 regarded as extensible structures and whose morphisms are the functions between their base sets. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 17833 we do not need to restrict the universe to sets which "have a base". 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. (Proposed by Mario Carneiro, 5-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
Assertion
Ref Expression
df-estrc ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})
Distinct variable group:   𝑓,𝑔,𝑢,𝑣,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-estrc
StepHypRef Expression
1 cestrc 17834 . 2 class ExtStrCat
2 vu . . 3 setvar 𝑢
3 cvv 3431 . . 3 class V
4 cnx 16890 . . . . . 6 class ndx
5 cbs 16908 . . . . . 6 class Base
64, 5cfv 6431 . . . . 5 class (Base‘ndx)
72cv 1541 . . . . 5 class 𝑢
86, 7cop 4573 . . . 4 class ⟨(Base‘ndx), 𝑢
9 chom 16969 . . . . . 6 class Hom
104, 9cfv 6431 . . . . 5 class (Hom ‘ndx)
11 vx . . . . . 6 setvar 𝑥
12 vy . . . . . 6 setvar 𝑦
1312cv 1541 . . . . . . . 8 class 𝑦
1413, 5cfv 6431 . . . . . . 7 class (Base‘𝑦)
1511cv 1541 . . . . . . . 8 class 𝑥
1615, 5cfv 6431 . . . . . . 7 class (Base‘𝑥)
17 cmap 8596 . . . . . . 7 class m
1814, 16, 17co 7269 . . . . . 6 class ((Base‘𝑦) ↑m (Base‘𝑥))
1911, 12, 7, 7, 18cmpo 7271 . . . . 5 class (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))
2010, 19cop 4573 . . . 4 class ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩
21 cco 16970 . . . . . 6 class comp
224, 21cfv 6431 . . . . 5 class (comp‘ndx)
23 vv . . . . . 6 setvar 𝑣
24 vz . . . . . 6 setvar 𝑧
257, 7cxp 5587 . . . . . 6 class (𝑢 × 𝑢)
26 vg . . . . . . 7 setvar 𝑔
27 vf . . . . . . 7 setvar 𝑓
2824cv 1541 . . . . . . . . 9 class 𝑧
2928, 5cfv 6431 . . . . . . . 8 class (Base‘𝑧)
3023cv 1541 . . . . . . . . . 10 class 𝑣
31 c2nd 7821 . . . . . . . . . 10 class 2nd
3230, 31cfv 6431 . . . . . . . . 9 class (2nd𝑣)
3332, 5cfv 6431 . . . . . . . 8 class (Base‘(2nd𝑣))
3429, 33, 17co 7269 . . . . . . 7 class ((Base‘𝑧) ↑m (Base‘(2nd𝑣)))
35 c1st 7820 . . . . . . . . . 10 class 1st
3630, 35cfv 6431 . . . . . . . . 9 class (1st𝑣)
3736, 5cfv 6431 . . . . . . . 8 class (Base‘(1st𝑣))
3833, 37, 17co 7269 . . . . . . 7 class ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣)))
3926cv 1541 . . . . . . . 8 class 𝑔
4027cv 1541 . . . . . . . 8 class 𝑓
4139, 40ccom 5593 . . . . . . 7 class (𝑔𝑓)
4226, 27, 34, 38, 41cmpo 7271 . . . . . 6 class (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))
4323, 24, 25, 7, 42cmpo 7271 . . . . 5 class (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))
4422, 43cop 4573 . . . 4 class ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩
458, 20, 44ctp 4571 . . 3 class {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩}
462, 3, 45cmpt 5162 . 2 class (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})
471, 46wceq 1542 1 wff ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})
Colors of variables: wff setvar class
This definition is referenced by:  estrcval  17836
  Copyright terms: Public domain W3C validator