P. 183 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	estrccat 18201	The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	estrcid 18202	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 18203	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 18204	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)
Theorem	estrreslem1 18205	Lemma 1 for estrres 18208. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	estrreslem1OLD 18206	Obsolete version of estrreslem1 18205 as of 28-Oct-2024. Lemma 1 for estrres 18208. (Contributed by AV, 14-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	estrreslem2 18207	Lemma 2 for estrres 18208. (Contributed by AV, 14-Mar-2020.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → · ∈ 𝑌)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐶)
Theorem	estrres 18208	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.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → · ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐶 ↾s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})
Theorem	funcestrcsetclem1 18209*	Lemma 1 for funcestrcsetc 18218. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
Theorem	funcestrcsetclem2 18210*	Lemma 2 for funcestrcsetc 18218. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcestrcsetclem3 18211*	Lemma 3 for funcestrcsetc 18218. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcestrcsetclem4 18212*	Lemma 4 for funcestrcsetc 18218. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcestrcsetclem5 18213*	Lemma 5 for funcestrcsetc 18218. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ 𝑀 = (Base‘𝑋)    &   ⊢ 𝑁 = (Base‘𝑌)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀)))
Theorem	funcestrcsetclem6 18214*	Lemma 6 for funcestrcsetc 18218. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ 𝑀 = (Base‘𝑋)    &   ⊢ 𝑁 = (Base‘𝑌)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcestrcsetclem7 18215*	Lemma 7 for funcestrcsetc 18218. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	funcestrcsetclem8 18216*	Lemma 8 for funcestrcsetc 18218. (Contributed by AV, 15-Feb-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))
Theorem	funcestrcsetclem9 18217*	Lemma 9 for funcestrcsetc 18218. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcestrcsetc 18218*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
Theorem	fthestrcsetc 18219*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
Theorem	fullestrcsetc 18220*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
Theorem	equivestrcsetc 18221*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎)))
Theorem	setc1strwun 18222	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ 𝑈)
Theorem	funcsetcestrclem1 18223*	Lemma 1 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})
Theorem	funcsetcestrclem2 18224*	Lemma 2 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcsetcestrclem3 18225*	Lemma 3 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)
Theorem	embedsetcestrclem 18226*	Lemma for embedsetcestrc 18236. (Contributed by AV, 31-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵)
Theorem	funcsetcestrclem4 18227*	Lemma 4 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶))
Theorem	funcsetcestrclem5 18228*	Lemma 5 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑m 𝑋)))
Theorem	funcsetcestrclem6 18229*	Lemma 6 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcsetcestrclem7 18230*	Lemma 7 for funcsetcestrc 18233. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
Theorem	funcsetcestrclem8 18231*	Lemma 8 for funcsetcestrc 18233. (Contributed by AV, 28-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑆)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
Theorem	funcsetcestrclem9 18232*	Lemma 9 for funcsetcestrc 18233. (Contributed by AV, 28-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcsetcestrc 18233*	The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺)
Theorem	fthsetcestrc 18234*	The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is faithful. (Contributed by AV, 31-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Faith 𝐸)𝐺)
Theorem	fullsetcestrc 18235*	The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺)
Theorem	embedsetcestrc 18236*	The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is an embedding. According to definition 3.27 (1) of [Adamek] p. 34, a functor "F is called an embedding provided that F is injective on morphisms", or according to remark 3.28 (1) in [Adamek] p. 34, "a functor is an embedding if and only if it is faithful and injective on objects". (Contributed by AV, 31-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → (𝐹(𝑆 Faith 𝐸)𝐺 ∧ 𝐹:𝐶–1-1→𝐵))
8.4  Categorical constructions
8.4.1  Product of categories
Syntax	cxpc 18237	Extend class notation with the product of two categories.
class ×c
Syntax	c1stf 18238	Extend class notation with the first projection functor.
class 1stF
Syntax	c2ndf 18239	Extend class notation with the second projection functor.
class 2ndF
Syntax	cprf 18240	Extend class notation with the functor pairing operation.
class ⟨,⟩F
Definition	df-xpc 18241*	Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
Definition	df-1stf 18242*	Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
Definition	df-2ndf 18243*	Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
Definition	df-prf 18244*	Define the pairing operation for functors (which takes two functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐸 and produces (𝐹 ⟨,⟩F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩)
Theorem	fnxpc 18245	The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ ×c Fn (V × V)
Theorem	xpcval 18246*	Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌))    &   ⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))    &   ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))    ⇒   ⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
Theorem	xpcbas 18247	Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    ⇒   ⊢ (𝑋 × 𝑌) = (Base‘𝑇)
Theorem	xpchomfval 18248*	Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
Theorem	xpchom 18249	Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))
Theorem	relxpchom 18250	A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ Rel (𝑋𝐾𝑌)
Theorem	xpccofval 18251*	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝑇)    ⇒   ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩))
Theorem	xpcco 18252	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩)
Theorem	xpcco1st 18253	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)))
Theorem	xpcco2nd 18254	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    &   ⊢ · = (comp‘𝐷)    ⇒   ⊢ (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)))
Theorem	xpchom2 18255	Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑌)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
Theorem	xpcco2 18256	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑌)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺)⟩)
Theorem	xpccatid 18257*	The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ 𝐽 = (Id‘𝐷)    ⇒   ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))
Theorem	xpcid 18258	The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ 𝐽 = (Id‘𝐷)    &   ⊢ 1 = (Id‘𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩)
Theorem	xpccat 18259	The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑇 ∈ Cat)
Theorem	1stfval 18260*	Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    ⇒   ⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
Theorem	1stf1 18261	Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = (1st ‘𝑅))
Theorem	1stf2 18262	Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))
Theorem	2ndfval 18263*	Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 2ndF 𝐷)    ⇒   ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
Theorem	2ndf1 18264	Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 2ndF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅))
Theorem	2ndf2 18265	Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 2ndF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))
Theorem	1stfcl 18266	The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    ⇒   ⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶))
Theorem	2ndfcl 18267	The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 2ndF 𝐷)    ⇒   ⊢ (𝜑 → 𝑄 ∈ (𝑇 Func 𝐷))
Theorem	prfval 18268*	Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    ⇒   ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ 𝐵 ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
Theorem	prf1 18269	Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = ⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)⟩)
Theorem	prf2fval 18270*	Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)⟩))
Theorem	prf2 18271	Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)⟩)
Theorem	prfcl 18272	The pairing of functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐷 is a functor ⟨𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ 𝑇 = (𝐷 ×c 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    ⇒   ⊢ (𝜑 → 𝑃 ∈ (𝐶 Func 𝑇))
Theorem	prf1st 18273	Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    ⇒   ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)
Theorem	prf2nd 18274	Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    ⇒   ⊢ (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)
Theorem	1st2ndprf 18275	Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑇 = (𝐷 ×c 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇))    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))
Theorem	catcxpccl 18276	The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑇 = (𝑋 ×c 𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐵)
Theorem	catcxpcclOLD 18277	Obsolete proof of catcxpccl 18276 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑇 = (𝑋 ×c 𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐵)
Theorem	xpcpropd 18278	If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))
8.4.2  Functor evaluation
Syntax	cevlf 18279	Extend class notation with the evaluation functor.
class evalF
Syntax	ccurf 18280	Extend class notation with the currying of a functor.
class curryF
Syntax	cuncf 18281	Extend class notation with the uncurrying of a functor.
class uncurryF
Syntax	cdiag 18282	Extend class notation to include the diagonal functor.
class Δfunc
Definition	df-evlf 18283*	Define the evaluation functor, which is the extension of the evaluation map 𝑓, 𝑥 ↦ (𝑓‘𝑥) of functors, to a functor (𝐶⟶𝐷) × 𝐶⟶𝐷. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩(comp‘𝑑)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩)
Definition	df-curf 18284*	Define the curry functor, which maps a functor 𝐹:𝐶 × 𝐷⟶𝐸 to curryF (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st ‘𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)
Definition	df-uncf 18285*	Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
Definition	df-diag 18286*	Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). The value of the functor at an object 𝑥 is the constant functor which maps all objects in 𝐷 to 𝑥 and all morphisms to 1(𝑥). The morphism part is a natural transformation between these functors, which takes 𝑓:𝑥⟶𝑦 to the natural transformation with every component equal to 𝑓. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (⟨𝑐, 𝑑⟩ curryF (𝑐 1stF 𝑑)))
Theorem	evlfval 18287*	Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    ⇒   ⊢ (𝜑 → 𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(⟨((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))⟩ · ((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))⟩)
Theorem	evlf2 18288*	Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐿 = (⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐺, 𝑌⟩)    ⇒   ⊢ (𝜑 → 𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔))))
Theorem	evlf2val 18289	Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐿 = (⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐺, 𝑌⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))
Theorem	evlf1 18290	Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋))
Theorem	evlfcllem 18291	Lemma for evlfcl 18292. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (Base‘𝐶)))    &   ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))    &   ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))    &   ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))    &   ⊢ (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))    ⇒   ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st ‘𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))
Theorem	evlfcl 18292	The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐶⟶𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐸 = (𝐶 evalF 𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))
Theorem	curfval 18293*	Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    ⇒   ⊢ (𝜑 → 𝐺 = ⟨(𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd ‘𝐹)⟨𝑦, 𝑧⟩)(𝐼‘𝑧)))))⟩)
Theorem	curf1fval 18294*	Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ ⟨(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(⟨𝑥, 𝑦⟩(2nd ‘𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))
Theorem	curf1 18295*	Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(⟨𝑋, 𝑦⟩(2nd ‘𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)
Theorem	curf11 18296	Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌))
Theorem	curf12 18297	The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))    ⇒   ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑋, 𝑍⟩)𝐻))
Theorem	curf1cl 18298	The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
Theorem	curf2 18299*	Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)    ⇒   ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd ‘𝐹)⟨𝑌, 𝑧⟩)(𝐼‘𝑧))))
Theorem	curf2val 18300	Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd ‘𝐹)⟨𝑌, 𝑍⟩)(𝐼‘𝑍)))