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Metamath Proof Explorer

Definition df-termo

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 and dftermo3 for alternate definitions depending on df-inito . See dftermo4 for an alternate definition using the universal property. (Contributed by AV, 3-Apr-2020)

Ref Expression
Assertion df-termo ⊢ TermO = c ∈ Cat ⟼ a ∈ Base c | ∀ b ∈ Base c ∃! h h ∈ b Hom ⁡ c a

Detailed syntax breakdown

Step Hyp Ref Expression
0 ctermo class TermO
1 vc setvar c
2 ccat class Cat
3 va setvar a
4 cbs class Base
5 1 cv setvar c
6 5 4 cfv class Base c
7 vb setvar b
8 vh setvar h
9 8 cv setvar h
10 7 cv setvar b
11 chom class Hom
12 5 11 cfv class Hom ⁡ c
13 3 cv setvar a
14 10 13 12 co class b Hom ⁡ c a
15 9 14 wcel wff h ∈ b Hom ⁡ c a
16 15 8 weu wff ∃! h h ∈ b Hom ⁡ c a
17 16 7 6 wral wff ∀ b ∈ Base c ∃! h h ∈ b Hom ⁡ c a
18 17 3 6 crab class a ∈ Base c | ∀ b ∈ Base c ∃! h h ∈ b Hom ⁡ c a
19 1 2 18 cmpt class c ∈ Cat ⟼ a ∈ Base c | ∀ b ∈ Base c ∃! h h ∈ b Hom ⁡ c a
20 0 19 wceq wff TermO = c ∈ Cat ⟼ a ∈ Base c | ∀ b ∈ Base c ∃! h h ∈ b Hom ⁡ c a