P. 181 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	homarel 18001	An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ Rel (𝑋𝐻𝑌)
Theorem	homa1 18002	The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = ⟨𝑋, 𝑌⟩)
Theorem	homahom2 18003	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))
Theorem	homahom 18004	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))
Theorem	homadm 18005	The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋)
Theorem	homacd 18006	The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
Theorem	homadmcd 18007	Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
Theorem	arwval 18008	The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ 𝐴 = ∪ ran 𝐻
Theorem	arwrcl 18009	The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)
Theorem	arwhoma 18010	An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
Theorem	homarw 18011	A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝑋𝐻𝑌) ⊆ 𝐴
Theorem	arwdm 18012	The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (doma‘𝐹) ∈ 𝐵)
Theorem	arwcd 18013	The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (coda‘𝐹) ∈ 𝐵)
Theorem	dmaf 18014	The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (doma ↾ 𝐴):𝐴⟶𝐵
Theorem	cdaf 18015	The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (coda ↾ 𝐴):𝐴⟶𝐵
Theorem	arwhom 18016	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)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹)))
Theorem	arwdmcd 18017	Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹 = ⟨(doma‘𝐹), (coda‘𝐹), (2nd ‘𝐹)⟩)
8.2.1  Identity and composition for arrows
Syntax	cida 18018	Extend class notation to include identity for arrows.
class Ida
Syntax	ccoa 18019	Extend class notation to include composition for arrows.
class compa
Definition	df-ida 18020*	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)⟩))
Definition	df-coa 18021*	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 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩))
Theorem	idafval 18022*	Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
Theorem	idaval 18023	Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩)
Theorem	ida2 18024	Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋))
Theorem	idahom 18025	Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋))
Theorem	idadm 18026	Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (doma‘(𝐼‘𝑋)) = 𝑋)
Theorem	idacd 18027	Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (coda‘(𝐼‘𝑋)) = 𝑋)
Theorem	idaf 18028	The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐼 = (Ida‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝜑 → 𝐼:𝐵⟶𝐴)
Theorem	coafval 18029*	The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ ∙ = (comp‘𝐶)    ⇒   ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)
Theorem	eldmcoa 18030	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.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺)))
Theorem	dmcoass 18031	The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ dom · ⊆ (𝐴 × 𝐴)
Theorem	homdmcoa 18032	If 𝐹:𝑋⟶𝑌 and 𝐺:𝑌⟶𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → 𝐺dom · 𝐹)
Theorem	coaval 18033	Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ ∙ = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)
Theorem	coa2 18034	The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ ∙ = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))
Theorem	coahom 18035	The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
Theorem	coapm 18036	Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ · = (compa‘𝐶)    &   ⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ · ∈ (𝐴 ↑pm (𝐴 × 𝐴))
Theorem	arwlid 18037	Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ · = (compa‘𝐶)    &   ⊢ 1 = (Ida‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹)
Theorem	arwrid 18038	Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ · = (compa‘𝐶)    &   ⊢ 1 = (Ida‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹)
Theorem	arwass 18039	Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ · = (compa‘𝐶)    &   ⊢ 1 = (Ida‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))    ⇒   ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
8.3  Examples of categories
8.3.1  The category of sets
Syntax	csetc 18040	Extend class notation to include the category Set.
class SetCat
Definition	df-setc 18041*	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), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	setcval 18042*	Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	setcbas 18043	Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
Theorem	setchomfval 18044*	Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥)))
Theorem	setchom 18045	Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑m 𝑋))
Theorem	elsetchom 18046	A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝑋⟶𝑌))
Theorem	setccofval 18047*	Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
Theorem	setcco 18048	Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑍)    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	setccatid 18049*	Lemma for setccat 18050. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥))))
Theorem	setccat 18050	The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	setcid 18051	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑋))
Theorem	setcmon 18052	A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌))
Theorem	setcepi 18053	An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ (𝜑 → 2o ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌))
Theorem	setcsect 18054	A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋))))
Theorem	setcinv 18055	An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐺 = ◡𝐹)))
Theorem	setciso 18056	An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌))
Theorem	resssetc 18057	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))
Theorem	funcsetcres2 18058	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → (𝐸 Func 𝐷) ⊆ (𝐸 Func 𝐶))
Theorem	setc2obas 18059	∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 18060 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17632 and cat1 18062. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ 𝐶 = (SetCat‘2o)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅)
Theorem	setc2ohom 18060	(SetCat‘2o) is a category (provable from setccat 18050 and 2oex 8413) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18059. Notably, the empty set ∅ is simultaneously an object (setc2obas 18059), an identity morphism from ∅ to ∅ (setcid 18051 or thincid 49929), and a non-identity morphism from ∅ to 1o. See cat1lem 18061 and cat1 18062 for a more general statement. This category is also thin (setc2othin 49963), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 49961 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ 𝐶 = (SetCat‘2o)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o))
Theorem	cat1lem 18061*	The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 18062. (Contributed by Zhi Wang, 15-Sep-2024.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → ∅ ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → ∅ ≠ 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	cat1 18062*	The definition of category df-cat 17632 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18059 and setc2ohom 18060 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17991 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
8.3.2  The category of categories
Syntax	ccatc 18063	Extend class notation to include the category Cat.
class CatCat
Definition	df-catc 18064*	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 (catchom 18068, elcatchom 49894). 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), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
Theorem	catcval 18065*	Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	catcbas 18066	Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
Theorem	catchomfval 18067*	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
Theorem	catchom 18068	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌))
Theorem	catccofval 18069*	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
Theorem	catcco 18070	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))
Theorem	catccatid 18071*	Lemma for catccat 18073. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (idfunc‘𝑥))))
Theorem	catcid 18072	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 = (idfunc‘𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = 𝐼)
Theorem	catccat 18073	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)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	resscatc 18074	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))
Theorem	catcisolem 18075*	Lemma for catciso 18076. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Inv‘𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡𝐹‘𝑥)𝐺(◡𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)    &   ⊢ (𝜑 → 𝐹:𝑅–1-1-onto→𝑆)    ⇒   ⊢ (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨◡𝐹, 𝐻⟩)
Theorem	catciso 18076	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. Note that "catciso.u" is redundant thanks to elbasfv 17183. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆)))
Theorem	catcbascl 18077	An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18082. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑈)
Theorem	catcslotelcl 18078	A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18082. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝑋) ∈ 𝑈)
Theorem	catcbaselcl 18079	The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18082. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
Theorem	catchomcl 18080	The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18082. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
Theorem	catcccocl 18081	The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18082. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
Theorem	catcoppccl 18082	The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝐵)
Theorem	catcfuccl 18083	The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑄 = (𝑋 FuncCat 𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑄 ∈ 𝐵)
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 18087. 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 18089. 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 18085 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 18090, whereas the composition is the ordinary composition of functions, see estrccofval 18093 and estrcco 18094. It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 18097 with the identity function as identity arrow, see estrcid 18098. 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 18098): At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordered pairs, see dfrngc2 20607 and dfringc2 20636, but with special compositions). With this definitions, however, Theorem rngcresringcat 20648 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 18041, 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 18087. The claimed equivalence is proven by equivestrcsetc 18116. 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 20596 and df-ringc 20625. In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 20648, 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 18116. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 18131. Remark: equivestrcsetc 18116 as well as embedsetcestrc 18131 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 17163), 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 17186. 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 20217) 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)⟩
Theorem	fncnvimaeqv 18084	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)
Theorem	bascnvimaeqv 18085	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
Syntax	cestrc 18086	Extend class notation to include the category ExtStr.
class ExtStrCat
Definition	df-estrc 18087*	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 18085 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), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	estrcval 18088*	Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	estrcbas 18089	Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
Theorem	estrchomfval 18090*	Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	estrchom 18091	The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝐵 ↑m 𝐴))
Theorem	elestrchom 18092	A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝐴⟶𝐵))
Theorem	estrccofval 18093*	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
Theorem	estrcco 18094	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐷 = (Base‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	estrcbasbas 18095	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    ⇒   ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → (Base‘𝐸) ∈ 𝑈)
Theorem	estrccatid 18096*	Lemma for estrccat 18097. (Contributed by AV, 8-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Theorem	estrccat 18097	The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	estrcid 18098	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ (Base‘𝑋)))
Theorem	estrchomfn 18099	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
Theorem	estrchomfeqhom 18100	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (Homf ‘𝐶) = 𝐻)