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Theorem List for Metamath Proof Explorer - 17301-17400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremarwhom 17301 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝐹𝐴 → (2nd𝐹) ∈ ((doma𝐹)𝐽(coda𝐹)))
 
Theoremarwdmcd 17302 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)       (𝐹𝐴𝐹 = ⟨(doma𝐹), (coda𝐹), (2nd𝐹)⟩)
 
8.2.1  Identity and composition for arrows
 
Syntaxcida 17303 Extend class notation to include identity for arrows.
class Ida
 
Syntaxccoa 17304 Extend class notation to include composition for arrows.
class compa
 
Definitiondf-ida 17305* Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
 
Definitiondf-coa 17306* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a quinary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
 
Theoremidafval 17307* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)       (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
 
Theoremidaval 17308 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
 
Theoremida2 17309 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
 
Theoremidahom 17310 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Homa𝐶)       (𝜑 → (𝐼𝑋) ∈ (𝑋𝐻𝑋))
 
Theoremidadm 17311 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (doma‘(𝐼𝑋)) = 𝑋)
 
Theoremidacd 17312 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (coda‘(𝐼𝑋)) = 𝑋)
 
Theoremidaf 17313 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐴 = (Arrow‘𝐶)       (𝜑𝐼:𝐵𝐴)
 
Theoremcoafval 17314* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)    &    = (comp‘𝐶)        · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
 
Theoremeldmcoa 17315 A pair 𝐺, 𝐹 is in the domain of the arrow composition, if the domain of 𝐺 equals the codomain of 𝐹. (In this case we say 𝐺 and 𝐹 are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)       (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
 
Theoremdmcoass 17316 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)       dom · ⊆ (𝐴 × 𝐴)
 
Theoremhomdmcoa 17317 If 𝐹:𝑋𝑌 and 𝐺:𝑌𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑𝐺dom · 𝐹)
 
Theoremcoaval 17318 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &    = (comp‘𝐶)       (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
 
Theoremcoa2 17319 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &    = (comp‘𝐶)       (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
 
Theoremcoahom 17320 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 17321 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 17322 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 17323 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 17324 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 17325 Extend class notation to include the category Set.
class SetCat
 
Definitiondf-setc 17326* 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 17327* 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 17328 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 17329* 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 17330 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 17331 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 17332* 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 17333 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 17334* Lemma for setccat 17335. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))
 
Theoremsetccat 17335 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)
 
Theoremsetcid 17336 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 17337 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 17338 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 17339 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 17340 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 17341 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 17342 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 17343 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 17344 Extend class notation to include the category Cat.
class CatCat
 
Definitiondf-catc 17345* Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e. "𝑢-small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "𝑢 -large categories", which can be expressed as (Cat ∖ 𝑢). (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 17346* 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 17347 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 17348* 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 17349 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 17350* 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 17351 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 17352* Lemma for catccat 17354. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))
 
Theoremcatcid 17353 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 17354 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 "𝑈 -large".) (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)
 
Theoremresscatc 17355 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 17356* Lemma for catciso 17357. (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 17357 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 17358 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 17359 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 17363.

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 17365. 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 17361 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 17366, whereas the composition is the ordinary composition of functions, see estrccofval 17369 and estrcco 17370.

It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 17373 with the identity function as identity arrow, see estrcid 17374.

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 17374):

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 44141 and dfringc2 44187, but with special compositions). With this definitions, however, theorem rngcresringcat 44199 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 17326, 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 17363. The claimed equivalence is proven by equivestrcsetc 17392. 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 44128 and df-ringc 44174.

In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 44199, 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 17392. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 17407.

Remark: equivestrcsetc 17392 as well as embedsetcestrc 17407 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 16494), 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 16537.

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) Ob(B) and MorA(X,Y) 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) 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 19233) 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), (+g‘ndx), (.r‘ndx)⟩

 
Theoremfncnvimaeqv 17360 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 17361 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 17362 Extend class notation to include the category ExtStr.
class ExtStrCat
 
Definitiondf-estrc 17363* 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 17361 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 17364* 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 17365 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 17366* 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 17367 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 17368 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 17369* 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 17370 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 17371 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 17372* Lemma for estrccat 17373. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))
 
Theoremestrccat 17373 The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)
 
Theoremestrcid 17374 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 17375 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 17376 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 17377 Lemma 1 for estrres 17379. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)       (𝜑𝐵 = (Base‘𝐶))
 
Theoremestrreslem2 17378 Lemma 2 for estrres 17379. (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 17379 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.) (Revised by AV, 3-Jul-2022.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)    &   (𝜑𝐺𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐶s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})
 
Theoremfuncestrcsetclem1 17380* Lemma 1 for funcestrcsetc 17389. (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 17381* Lemma 2 for funcestrcsetc 17389. (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 17382* Lemma 3 for funcestrcsetc 17389. (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 17383* Lemma 4 for funcestrcsetc 17389. (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 17384* Lemma 5 for funcestrcsetc 17389. (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 17385* Lemma 6 for funcestrcsetc 17389. (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 17386* Lemma 7 for funcestrcsetc 17389. (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 17387* Lemma 8 for funcestrcsetc 17389. (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 17388* Lemma 9 for funcestrcsetc 17389. (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 17389* 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 17390* 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 17391* 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 17392* 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 17393 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 17394* Lemma 1 for funcsetcestrc 17404. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))       ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
 
Theoremfuncsetcestrclem2 17395* Lemma 2 for funcsetcestrc 17404. (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 17396* Lemma 3 for funcsetcestrc 17404. (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 17397* Lemma for embedsetcestrc 17407. (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 17398* Lemma 4 for funcsetcestrc 17404. (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 17399* Lemma 5 for funcsetcestrc 17404. (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 17400* Lemma 6 for funcsetcestrc 17404. (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)
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