Definition df-termo 18037

Description: An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). See dftermo2 18056 and dftermo3 18058 for alternate definitions depending on df-inito 18036. See dftermo4 50300 for an alternate definition using the universal property. (Contributed by AV, 3-Apr-2020.)

Assertion

Ref	Expression
df-termo	⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})

Distinct variable group:   𝑎,𝑏,𝑐,ℎ

Detailed syntax breakdown of Definition df-termo

Step	Hyp	Ref	Expression
1		ctermo 18034	. 2 class TermO
2		vc	. . 3 setvar 𝑐
3		ccat 17715	. . 3 class Cat
4		vh	. . . . . . . 8 setvar ℎ
5	4	cv 1569	. . . . . . 7 class ℎ
6		vb	. . . . . . . . 9 setvar 𝑏
7	6	cv 1569	. . . . . . . 8 class 𝑏
8		va	. . . . . . . . 9 setvar 𝑎
9	8	cv 1569	. . . . . . . 8 class 𝑎
10	2	cv 1569	. . . . . . . . 9 class 𝑐
11		chom 17316	. . . . . . . . 9 class Hom
12	10, 11	cfv 6536	. . . . . . . 8 class (Hom ‘𝑐)
13	7, 9, 12	co 7410	. . . . . . 7 class (𝑏(Hom ‘𝑐)𝑎)
14	5, 13	wcel 2143	. . . . . 6 wff ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)
15	14, 4	weu 2596	. . . . 5 wff ∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)
16		cbs 17264	. . . . . 6 class Base
17	10, 16	cfv 6536	. . . . 5 class (Base‘𝑐)
18	15, 6, 17	wral 3079	. . . 4 wff ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)
19	18, 8, 17	crab 3416	. . 3 class {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}
20	2, 3, 19	cmpt 5192	. 2 class (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
21	1, 20	wceq 1570	1 wff TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})

This definition is referenced by:  termofn  18040  termorcl  18043  termoval  18046  dftermo2  18056