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Definition df-full 16333
Description: Function returning all the full functors from a category 𝐶 to a category 𝐷. A full functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌𝐶 are surjections. Definition 3.27(3) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
df-full Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
Distinct variable group:   𝑐,𝑑,𝑓,𝑔,𝑥,𝑦

Detailed syntax breakdown of Definition df-full
StepHypRef Expression
1 cful 16331 . 2 class Full
2 vc . . 3 setvar 𝑐
3 vd . . 3 setvar 𝑑
4 ccat 16094 . . 3 class Cat
5 vf . . . . . . 7 setvar 𝑓
65cv 1473 . . . . . 6 class 𝑓
7 vg . . . . . . 7 setvar 𝑔
87cv 1473 . . . . . 6 class 𝑔
92cv 1473 . . . . . . 7 class 𝑐
103cv 1473 . . . . . . 7 class 𝑑
11 cfunc 16283 . . . . . . 7 class Func
129, 10, 11co 6527 . . . . . 6 class (𝑐 Func 𝑑)
136, 8, 12wbr 4577 . . . . 5 wff 𝑓(𝑐 Func 𝑑)𝑔
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1473 . . . . . . . . . 10 class 𝑥
16 vy . . . . . . . . . . 11 setvar 𝑦
1716cv 1473 . . . . . . . . . 10 class 𝑦
1815, 17, 8co 6527 . . . . . . . . 9 class (𝑥𝑔𝑦)
1918crn 5029 . . . . . . . 8 class ran (𝑥𝑔𝑦)
2015, 6cfv 5790 . . . . . . . . 9 class (𝑓𝑥)
2117, 6cfv 5790 . . . . . . . . 9 class (𝑓𝑦)
22 chom 15725 . . . . . . . . . 10 class Hom
2310, 22cfv 5790 . . . . . . . . 9 class (Hom ‘𝑑)
2420, 21, 23co 6527 . . . . . . . 8 class ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))
2519, 24wceq 1474 . . . . . . 7 wff ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))
26 cbs 15641 . . . . . . . 8 class Base
279, 26cfv 5790 . . . . . . 7 class (Base‘𝑐)
2825, 16, 27wral 2895 . . . . . 6 wff 𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))
2928, 14, 27wral 2895 . . . . 5 wff 𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))
3013, 29wa 382 . . . 4 wff (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))
3130, 5, 7copab 4636 . . 3 class {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))}
322, 3, 4, 4, 31cmpt2 6529 . 2 class (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
331, 32wceq 1474 1 wff Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
Colors of variables: wff setvar class
This definition is referenced by:  fullfunc  16335  isfull  16339
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