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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhomarcl 16501 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Theoremhomafval 16502* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))

Theoremhomaf 16503 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Theoremhomaval 16504 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Theoremelhoma 16505 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))

Theoremelhomai 16506 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Theoremelhomai2 16507 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Theoremhomarcl2 16508 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋𝐵𝑌𝐵))

Theoremhomarel 16509 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       Rel (𝑋𝐻𝑌)

Theoremhoma1 16510 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝑍 = ⟨𝑋, 𝑌⟩)

Theoremhomahom2 16511 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝐹 ∈ (𝑋𝐽𝑌))

Theoremhomahom 16512 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋𝐽𝑌))

Theoremhomadm 16513 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)

Theoremhomacd 16514 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)

Theoremhomadmcd 16515 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)

Theoremarwval 16516 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       𝐴 = ran 𝐻

Theoremarwrcl 16517 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)       (𝐹𝐴𝐶 ∈ Cat)

Theoremarwhoma 16518 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 = (Homa𝐶)       (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))

Theoremhomarw 16519 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       (𝑋𝐻𝑌) ⊆ 𝐴

Theoremarwdm 16520 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (doma𝐹) ∈ 𝐵)

Theoremarwcd 16521 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (coda𝐹) ∈ 𝐵)

Theoremdmaf 16522 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (doma𝐴):𝐴𝐵

Theoremcdaf 16523 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (coda𝐴):𝐴𝐵

Theoremarwhom 16524 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 16525 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 16526 Extend class notation to include identity for arrows.
class Ida

Syntaxccoa 16527 Extend class notation to include composition for arrows.
class compa

Definitiondf-ida 16528* 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 16529* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a 5-ary 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 16530* 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 16531 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 16532 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 16533 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 16534 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 16535 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 16536 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 16537* 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 16538 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 16539 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 16540 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 16541 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 16542 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 16543 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 16544 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 16545 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 16546 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 16547 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 16548 Extend class notation to include the category Set.
class SetCat

Definitiondf-setc 16549* 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 16550* 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 16551 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 16552* 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 16553 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 16554 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 16555* 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 16556 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 16557* Lemma for setccat 16558. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))

Theoremsetccat 16558 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremsetcid 16559 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 16560 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 16561 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 16562 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 16563 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 16564 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 16565 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 16566 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 16567 Extend class notation to include the category Cat.
class CatCat

Definitiondf-catc 16568* 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 16569* 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 16570 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 16571* 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 16572 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 16573* 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 16574 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 16575* Lemma for catccat 16577. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))

Theoremcatcid 16576 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 16577 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 16578 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 16579* Lemma for catciso 16580. (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 16580 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 16581 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 16582 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 16586.

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 16588. 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 16584 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 16589, whereas the composition is the ordinary composition of functions, see estrccofval 16592 and estrcco 16593.

It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 16596 with the identity function as identity arrow, see estrcid 16597.

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 41764 and dfringc2 41810, but with special compositions). With this definitions, however, theorem rngcresringcat 41822 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 16549, 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 16586. The claimed equivalence is proven by equivestrcsetc 16615. 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 41751 and df-ringc 41797.

In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 41822, 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 16615. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 16630.

Remark: equivestrcsetc 16615 as well as embedsetcestrc 16630 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 15711), 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 15751.

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 18375) 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 16583 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 16584 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 16585 Extend class notation to include the category ExtStr.
class ExtStrCat

Definitiondf-estrc 16586* 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 16584 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 16587* 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 16588 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 16589* 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 16590 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 16591 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 16592* 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 16593 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 16594 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 16595* Lemma for estrccat 16596. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))

Theoremestrccat 16596 The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremestrcid 16597 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 16598 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 16599 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 16600 Lemma 1 for estrres 16602. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)       (𝜑𝐵 = (Base‘𝐶))

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