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Definition df-estrc 16810
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 16808 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 16809 . 2 class ExtStrCat
2 vu . . 3 setvar 𝑢
3 cvv 3231 . . 3 class V
4 cnx 15901 . . . . . 6 class ndx
5 cbs 15904 . . . . . 6 class Base
64, 5cfv 5926 . . . . 5 class (Base‘ndx)
72cv 1522 . . . . 5 class 𝑢
86, 7cop 4216 . . . 4 class ⟨(Base‘ndx), 𝑢
9 chom 15999 . . . . . 6 class Hom
104, 9cfv 5926 . . . . 5 class (Hom ‘ndx)
11 vx . . . . . 6 setvar 𝑥
12 vy . . . . . 6 setvar 𝑦
1312cv 1522 . . . . . . . 8 class 𝑦
1413, 5cfv 5926 . . . . . . 7 class (Base‘𝑦)
1511cv 1522 . . . . . . . 8 class 𝑥
1615, 5cfv 5926 . . . . . . 7 class (Base‘𝑥)
17 cmap 7899 . . . . . . 7 class 𝑚
1814, 16, 17co 6690 . . . . . 6 class ((Base‘𝑦) ↑𝑚 (Base‘𝑥))
1911, 12, 7, 7, 18cmpt2 6692 . . . . 5 class (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
2010, 19cop 4216 . . . 4 class ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩
21 cco 16000 . . . . . 6 class comp
224, 21cfv 5926 . . . . 5 class (comp‘ndx)
23 vv . . . . . 6 setvar 𝑣
24 vz . . . . . 6 setvar 𝑧
257, 7cxp 5141 . . . . . 6 class (𝑢 × 𝑢)
26 vg . . . . . . 7 setvar 𝑔
27 vf . . . . . . 7 setvar 𝑓
2824cv 1522 . . . . . . . . 9 class 𝑧
2928, 5cfv 5926 . . . . . . . 8 class (Base‘𝑧)
3023cv 1522 . . . . . . . . . 10 class 𝑣
31 c2nd 7209 . . . . . . . . . 10 class 2nd
3230, 31cfv 5926 . . . . . . . . 9 class (2nd𝑣)
3332, 5cfv 5926 . . . . . . . 8 class (Base‘(2nd𝑣))
3429, 33, 17co 6690 . . . . . . 7 class ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣)))
35 c1st 7208 . . . . . . . . . 10 class 1st
3630, 35cfv 5926 . . . . . . . . 9 class (1st𝑣)
3736, 5cfv 5926 . . . . . . . 8 class (Base‘(1st𝑣))
3833, 37, 17co 6690 . . . . . . 7 class ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣)))
3926cv 1522 . . . . . . . 8 class 𝑔
4027cv 1522 . . . . . . . 8 class 𝑓
4139, 40ccom 5147 . . . . . . 7 class (𝑔𝑓)
4226, 27, 34, 38, 41cmpt2 6692 . . . . . 6 class (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))
4323, 24, 25, 7, 42cmpt2 6692 . . . . 5 class (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓)))
4422, 43cop 4216 . . . 4 class ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩
458, 20, 44ctp 4214 . . 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 4762 . 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 1523 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  16811
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