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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcoahom 16701 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Theoremcoapm 16702 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)        · ∈ (𝐴pm (𝐴 × 𝐴))

Theoremarwlid 16703 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)

Theoremarwrid 16704 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)

Theoremarwass 16705 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑𝐾 ∈ (𝑍𝐻𝑊))       (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

8.3  Examples of categories

8.3.1  The category of sets

Syntaxcsetc 16706 Extend class notation to include the category Set.
class SetCat

Definitiondf-setc 16707* Definition of the category Set, relativized to a subset 𝑢. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in 𝑢 and functions between these sets. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩})

Theoremsetcval 16708* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))    &   (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremsetcbas 16709 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)       (𝜑𝑈 = (Base‘𝐶))

Theoremsetchomfval 16710* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))

Theoremsetchom 16711 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋𝐻𝑌) = (𝑌𝑚 𝑋))

Theoremelsetchom 16712 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝑋𝑌))

Theoremsetccofval 16713* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))

Theoremsetcco 16714 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   (𝜑𝐹:𝑋𝑌)    &   (𝜑𝐺:𝑌𝑍)       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremsetccatid 16715* Lemma for setccat 16716. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))

Theoremsetccat 16716 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremsetcid 16717 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑋))

Theoremsetcmon 16718 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑀 = (Mono‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))

Theoremsetcepi 16719 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐸 = (Epi‘𝐶)    &   (𝜑 → 2𝑜𝑈)       (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))

Theoremsetcsect 16720 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋𝑌𝐺:𝑌𝑋 ∧ (𝐺𝐹) = ( I ↾ 𝑋))))

Theoremsetcinv 16721 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹:𝑋1-1-onto𝑌𝐺 = 𝐹)))

Theoremsetciso 16722 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋1-1-onto𝑌))

Theoremresssetc 16723 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   𝐷 = (SetCat‘𝑉)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Theoremfuncsetcres2 16724 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   𝐷 = (SetCat‘𝑉)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → (𝐸 Func 𝐷) ⊆ (𝐸 Func 𝐶))

8.3.2  The category of categories

Syntaxccatc 16725 Extend class notation to include the category Cat.
class CatCat

Definitiondf-catc 16726* Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e. "small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat = (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})

Theoremcatcval 16727* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Cat))    &   (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))    &   (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremcatcbas 16728 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Cat))

Theoremcatchomfval 16729* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))

Theoremcatchom 16730 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌))

Theoremcatccofval 16731* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))

Theoremcatcco 16732 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 Func 𝑌))    &   (𝜑𝐺 ∈ (𝑌 Func 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺func 𝐹))

Theoremcatccatid 16733* Lemma for catccat 16735. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))

Theoremcatcid 16734 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝐼 = (idfunc𝑋)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) = 𝐼)

Theoremcatccat 16735 The category of categories is a category, see remark 3.48 in [Adamek] p. 40. (Clearly it cannot be an element of itself, hence it is "large" with respect to 𝑈.) (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremresscatc 16736 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐷 = (CatCat‘𝑉)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Theoremcatcisolem 16737* Lemma for catciso 16738. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Inv‘𝐶)    &   𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))    &   (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)    &   (𝜑𝐹:𝑅1-1-onto𝑆)       (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)

Theoremcatciso 16738 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)))

Theoremcatcoppccl 16739 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝑋)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑋𝐵)       (𝜑𝑂𝐵)

Theoremcatcfuccl 16740 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑄 = (𝑋 FuncCat 𝑌)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑𝑄𝐵)

8.3.3  The category of extensible structures

The "category of extensible structures" ExtStrCat is the category of all sets in a universe regarded as extensible structures and the functions between their base sets, see df-estrc 16744.

Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are all sets in a universe 𝑢, which can be an arbitrary set, see estrcbas 16746. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 16742 we do not need to restrict the universe to sets which "have a base". The morphisms (or arrows) between two objects, i.e. sets from the universe, are the mappings between their base sets, see estrchomfval 16747, whereas the composition is the ordinary composition of functions, see estrccofval 16750 and estrcco 16751.

It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 16754 with the identity function as identity arrow, see estrcid 16755.

In the following, some background information about the category of extensible structures is given, taken from the discussion in Github issue #1507 (see https://github.com/metamath/set.mm/issues/1507):

At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordereds pairs, see dfrngc2 41737 and dfringc2 41783, but with special compositions). With this definitions, however, theorem rngcresringcat 41795 could not be proven, because the compositions were not compatible. Unfortunately, no precise definition of the composition within the category of rings could be found in the literature. In section 3.3 EXAMPLES, paragraph (2) of [Adamek] p. 22, however, a definition is given for "Grp", the category of groups: "The following constructs; i.e., categories of structured sets and structure-preserving functions between them (o will always be the composition of functions and idA will always be the identity function on A): ... (b) Grp with objects all groups and morphisms all homomorphisms between them." Therefore, the compositions should have been harmonized by using the composition of the category of sets SetCat, see df-setc 16707, which is the ordinary composition of functions. Analogously, categories of Rngs (and Rings) could have been shown to be restrictions resp. subcategories of the category of sets.

BJ and MC observed, however, that "... cat [cannot be used] to restrict the category Set to Ring, because the homs are different. Although Ring is a concrete category, a hom between rings R and S is a function (Base`R) --> (Base`S) with certain properties, unlike in Set where it is a function R --> S.". Therefore, MC suggested that "we could have an alternative version of the Set category consisting of extensible structures (in U) together with (A Hom B) := (Base`A) --> (Base`B). This category is not isomorphic to Set because different extensible structures can have the same base set, but it is equivalent to Set; the relevant functors are (U`A) = (Base`A), the forgetful functor, and (F`A) = { <. (Base`ndx), A >. }". This led to the current definition of ExtStrCat, see df-estrc 16744. The claimed equivalence is proven by equivestrcsetc 16773. Having a definition of a category of extensible structures, the categories of non-unital and unital rings can be defined as appropriate restrictions of the category of extensible structures, see df-rngc 41724 and df-ringc 41770.

In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 41795, although the morphisms of the shown categories are different ( "->" means "is subcategory of"):

RingCat-> RngCat-> GrpCat -> MndCat -> MgmCat -> ExtStrCat

According to MC, "If we generalize from subcategories to embeddings, then we can even fit SetCat into the chain, equivalent to ExtStrCat at the end." As mentioned before, the equivalence of SetCat and ExtStrCat is proven by equivestrcsetc 16773. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 16788.

Remark: equivestrcsetc 16773 as well as embedsetcestrc 16788 require that the index of the base set extractor is contained within the considered universe. This is ensured by assuming that the natural numbers are contained within the considered universe: ω ∈ 𝑈 (see wunndx 15859), but it would be currently sufficient to assume that 1 ∈ 𝑈, because the index value of the base set extractor is hard-coded as 1, see basendx 15904.

Some people, however, feel uncomfortable to say that a ring "is a" group (without mentioning the restriction to the addition, which is usually found in the literature, e.g. the definition of a ring in [Herstein] p. 126: "... Note that so far all we have said is that R is an abelian group under +.". The main argument against a ring being a group is the number of components/slots: usually, a group consists of (exactly!) two components (a base set and an operation), whereas a ring consists of (exactly!) three components (a base set and two operations). According to this "definition", a ring cannot be a group.

This is also an (unfortunately informal) argument for the category of rings not being a subcategory of the category of abelian groups in "Categories and Functors", Bodo Pareigis, Academic Press, New York, London, 1970: "A category A is called a subcategory of a category B if Ob(A) C_ Ob(B) and MorA(X,Y) C_ MorB(X,Y) for all X,Y e. Ob(A), if the composition of morphisms in A coincides with the composition of the same morphisms in B and if the identity of an object in A is also the identity of the same object viewed as an object in B. Then there is a forgetful functor from A to B. We note that Ri [the category of rings] is not a subcategory of Ab [the category of abelian groups]. In fact, Ob(Ri) C_ Ob(Ab) is not true, although every ring can also be regarded as an abelian group. The corresponding abelian groups of two rings may coincide even if the rings do not coincide. The multiplication may be defined differently.".

As long as we define Rings, Groups, etc. in a way that 𝐴 ∈ Ring → 𝐴 ∈ Grp is valid (see ringgrp 18533) the corresponding categories are in a subcategory relation. If we do not want Rings to be Groups (then the category of rings would not be a subcategory of the category of groups, as observed by Pareigis), we would have to change the definitions of Magmas, Monoids, Groups, Rings etc. to restrict them to have exactly the required number of slots, so that the following holds

𝑔 ∈ Grp → 𝑔 Struct ⟨(Base‘ndx), (+g‘ndx)⟩

𝑟 ∈ Ring → 𝑟 Struct ⟨(Base‘ndx), (+gndx ) , ( .r ndx)⟩

Theoremfncnvimaeqv 16741 The inverse images of the universal class V under functions on the universal class V are the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
(𝐹 Fn V → (𝐹 “ V) = V)

Theorembascnvimaeqv 16742 The inverse image of the universal class V under the base function is the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
(Base “ V) = V

Syntaxcestrc 16743 Extend class notation to include the category ExtStr.
class ExtStrCat

Definitiondf-estrc 16744* Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe 𝑢 regarded as extensible structures and whose morphisms are the functions between their base sets. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 16742 we do not need to restrict the universe to sets which "have a base". Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Proposed by Mario Carneiro, 5-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})

Theoremestrcval 16745* Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))    &   (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremestrcbas 16746 Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)       (𝜑𝑈 = (Base‘𝐶))

Theoremestrchomfval 16747* Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremestrchom 16748 The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)       (𝜑 → (𝑋𝐻𝑌) = (𝐵𝑚 𝐴))

Theoremelestrchom 16749 A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝐴𝐵))

Theoremestrccofval 16750* Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))

Theoremestrcco 16751 Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)    &   𝐷 = (Base‘𝑍)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐷)       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremestrcbasbas 16752 An element of the base set of the base set of the category of extensible structures (i.e. the base set of an extensible structure) belongs to the considered weak universe. (Contributed by AV, 22-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝐸𝐵) → (Base‘𝐸) ∈ 𝑈)

Theoremestrccatid 16753* Lemma for estrccat 16754. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))

Theoremestrccat 16754 The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremestrcid 16755 The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)       (𝜑 → ( 1𝑋) = ( I ↾ (Base‘𝑋)))

Theoremestrchomfn 16756 The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 Fn (𝑈 × 𝑈))

Theoremestrchomfeqhom 16757 The functionalized Hom-set operation equals the Hom-set operation in the category of extensible structures (in a universe). (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (Homf𝐶) = 𝐻)

Theoremestrreslem1 16758 Lemma 1 for estrres 16760. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)       (𝜑𝐵 = (Base‘𝐶))

Theoremestrreslem2 16759 Lemma 2 for estrres 16760. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)       (𝜑 → (Base‘ndx) ∈ dom 𝐶)

Theoremestrres 16760 Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)    &   (𝜑𝐴𝑈)    &   (𝜑𝐺𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐶s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})

Theoremfuncestrcsetclem1 16761* Lemma 1 for funcestrcsetc 16770. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))

Theoremfuncestrcsetclem2 16762* Lemma 2 for funcestrcsetc 16770. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)

Theoremfuncestrcsetclem3 16763* Lemma 3 for funcestrcsetc 16770. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)

Theoremfuncestrcsetclem4 16764* Lemma 4 for funcestrcsetc 16770. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐺 Fn (𝐵 × 𝐵))

Theoremfuncestrcsetclem5 16765* Lemma 5 for funcestrcsetc 16770. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   𝑀 = (Base‘𝑋)    &   𝑁 = (Base‘𝑌)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))

Theoremfuncestrcsetclem6 16766* Lemma 6 for funcestrcsetc 16770. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   𝑀 = (Base‘𝑋)    &   𝑁 = (Base‘𝑌)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Theoremfuncestrcsetclem7 16767* Lemma 7 for funcestrcsetc 16770. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))

Theoremfuncestrcsetclem8 16768* Lemma 8 for funcestrcsetc 16770. (Contributed by AV, 15-Feb-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))

Theoremfuncestrcsetclem9 16769* Lemma 9 for funcestrcsetc 16770. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

Theoremfuncestrcsetc 16770* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Func 𝑆)𝐺)

Theoremfthestrcsetc 16771* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is faithful. (Contributed by AV, 2-Apr-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Faith 𝑆)𝐺)

Theoremfullestrcsetc 16772* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Full 𝑆)𝐺)

Theoremequivestrcsetc 16773* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (Contributed by AV, 2-Apr-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   (𝜑 → (Base‘ndx) ∈ 𝑈)       (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎)))

Theoremsetc1strwun 16774 A constructed one-slot structure with the objects of the category of sets as base set in a weak universe. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ 𝑈)

Theoremfuncsetcestrclem1 16775* Lemma 1 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))       ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})

Theoremfuncsetcestrclem2 16776* Lemma 2 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ 𝑈)

Theoremfuncsetcestrclem3 16777* Lemma 3 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑𝐹:𝐶𝐵)

Theoremembedsetcestrclem 16778* Lemma for embedsetcestrc 16788. (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑𝐹:𝐶1-1𝐵)

Theoremfuncsetcestrclem4 16779* Lemma 4 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       (𝜑𝐺 Fn (𝐶 × 𝐶))

Theoremfuncsetcestrclem5 16780* Lemma 5 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌𝑚 𝑋)))

Theoremfuncsetcestrclem6 16781* Lemma 6 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Theoremfuncsetcestrclem7 16782* Lemma 7 for funcsetcestrc 16785. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑𝑋𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))

Theoremfuncsetcestrclem8 16783* Lemma 8 for funcsetcestrc 16785. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑆)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))

Theoremfuncsetcestrclem9 16784* Lemma 9 for funcsetcestrc 16785. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

Theoremfuncsetcestrc 16785* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Func 𝐸)𝐺)

Theoremfthsetcestrc 16786* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is faithful. (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Faith 𝐸)𝐺)

Theoremfullsetcestrc 16787* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Full 𝐸)𝐺)

Theoremembedsetcestrc 16788* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is an embedding. According to definition 3.27 (1) of [Adamek] p. 34, a functor "F is called an embedding provided that F is injective on morphisms", or according to remark 3.28 (1) in [Adamek] p. 34, "a functor is an embedding if and only if it is faithful and injective on objects". (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑 → (𝐹(𝑆 Faith 𝐸)𝐺𝐹:𝐶1-1𝐵))

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 16789 Extend class notation with the product of two categories.
class ×c

Syntaxc1stf 16790 Extend class notation with the first projection functor.
class 1stF

Syntaxc2ndf 16791 Extend class notation with the second projection functor.
class 2ndF

Syntaxcprf 16792 Extend class notation with the functor pairing operation.
class ⟨,⟩F

Definitiondf-xpc 16793* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})

Definitiondf-1stf 16794* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)

Definitiondf-2ndf 16795* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)

Definitiondf-prf 16796* Define the pairing operation for functors (which takes two functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐸 and produces (𝐹 ⟨,⟩F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩)

Theoremfnxpc 16797 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
×c Fn (V × V)

Theoremxpcval 16798* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐵 = (𝑋 × 𝑌))    &   (𝜑𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))    &   (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))       (𝜑𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Theoremxpcbas 16799 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)       (𝑋 × 𝑌) = (Base‘𝑇)

Theoremxpchomfval 16800* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐾 = (Hom ‘𝑇)       𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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