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Definition df-fth 16612
Description: Function returning all the faithful functors from a category 𝐶 to a category 𝐷. A faithful functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌𝐶 are injections. Definition 3.27(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
df-fth Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
Distinct variable group:   𝑐,𝑑,𝑓,𝑔,𝑥,𝑦

Detailed syntax breakdown of Definition df-fth
StepHypRef Expression
1 cfth 16610 . 2 class Faith
2 vc . . 3 setvar 𝑐
3 vd . . 3 setvar 𝑑
4 ccat 16372 . . 3 class Cat
5 vf . . . . . . 7 setvar 𝑓
65cv 1522 . . . . . 6 class 𝑓
7 vg . . . . . . 7 setvar 𝑔
87cv 1522 . . . . . 6 class 𝑔
92cv 1522 . . . . . . 7 class 𝑐
103cv 1522 . . . . . . 7 class 𝑑
11 cfunc 16561 . . . . . . 7 class Func
129, 10, 11co 6690 . . . . . 6 class (𝑐 Func 𝑑)
136, 8, 12wbr 4685 . . . . 5 wff 𝑓(𝑐 Func 𝑑)𝑔
14 vx . . . . . . . . . . 11 setvar 𝑥
1514cv 1522 . . . . . . . . . 10 class 𝑥
16 vy . . . . . . . . . . 11 setvar 𝑦
1716cv 1522 . . . . . . . . . 10 class 𝑦
1815, 17, 8co 6690 . . . . . . . . 9 class (𝑥𝑔𝑦)
1918ccnv 5142 . . . . . . . 8 class (𝑥𝑔𝑦)
2019wfun 5920 . . . . . . 7 wff Fun (𝑥𝑔𝑦)
21 cbs 15904 . . . . . . . 8 class Base
229, 21cfv 5926 . . . . . . 7 class (Base‘𝑐)
2320, 16, 22wral 2941 . . . . . 6 wff 𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦)
2423, 14, 22wral 2941 . . . . 5 wff 𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦)
2513, 24wa 383 . . . 4 wff (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))
2625, 5, 7copab 4745 . . 3 class {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))}
272, 3, 4, 4, 26cmpt2 6692 . 2 class (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
281, 27wceq 1523 1 wff Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
Colors of variables: wff setvar class
This definition is referenced by:  fthfunc  16614  isfth  16621
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