Definition df-fth 17920

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, 𝑌 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		cfth 17918	. 2 class Faith
2		vc	. . 3 setvar 𝑐
3		vd	. . 3 setvar 𝑑
4		ccat 17676	. . 3 class Cat
5		vf	. . . . . . 7 setvar 𝑓
6	5	cv 1539	. . . . . 6 class 𝑓
7		vg	. . . . . . 7 setvar 𝑔
8	7	cv 1539	. . . . . 6 class 𝑔
9	2	cv 1539	. . . . . . 7 class 𝑐
10	3	cv 1539	. . . . . . 7 class 𝑑
11		cfunc 17867	. . . . . . 7 class Func
12	9, 10, 11	co 7405	. . . . . 6 class (𝑐 Func 𝑑)
13	6, 8, 12	wbr 5119	. . . . 5 wff 𝑓(𝑐 Func 𝑑)𝑔
14		vx	. . . . . . . . . . 11 setvar 𝑥
15	14	cv 1539	. . . . . . . . . 10 class 𝑥
16		vy	. . . . . . . . . . 11 setvar 𝑦
17	16	cv 1539	. . . . . . . . . 10 class 𝑦
18	15, 17, 8	co 7405	. . . . . . . . 9 class (𝑥𝑔𝑦)
19	18	ccnv 5653	. . . . . . . 8 class ◡(𝑥𝑔𝑦)
20	19	wfun 6525	. . . . . . 7 wff Fun ◡(𝑥𝑔𝑦)
21		cbs 17228	. . . . . . . 8 class Base
22	9, 21	cfv 6531	. . . . . . 7 class (Base‘𝑐)
23	20, 16, 22	wral 3051	. . . . . 6 wff ∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)
24	23, 14, 22	wral 3051	. . . . 5 wff ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)
25	13, 24	wa 395	. . . 4 wff (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))
26	25, 5, 7	copab 5181	. . 3 class {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))}
27	2, 3, 4, 4, 26	cmpo 7407	. 2 class (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
28	1, 27	wceq 1540	1 wff Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})

This definition is referenced by:  fthfunc  17922  isfth  17929