P. 182 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	estrreslem1 18101	Lemma 1 for estrres 18103. (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	estrreslem2 18102	Lemma 2 for estrres 18103. (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 18103	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 18104*	Lemma 1 for funcestrcsetc 18113. (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 18105*	Lemma 2 for funcestrcsetc 18113. (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 18106*	Lemma 3 for funcestrcsetc 18113. (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 18107*	Lemma 4 for funcestrcsetc 18113. (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 18108*	Lemma 5 for funcestrcsetc 18113. (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 18109*	Lemma 6 for funcestrcsetc 18113. (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 18110*	Lemma 7 for funcestrcsetc 18113. (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 18111*	Lemma 8 for funcestrcsetc 18113. (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 18112*	Lemma 9 for funcestrcsetc 18113. (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 18113*	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 18114*	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 18115*	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 18116*	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 18117	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 18118*	Lemma 1 for funcsetcestrc 18128. (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 18119*	Lemma 2 for funcsetcestrc 18128. (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 18120*	Lemma 3 for funcsetcestrc 18128. (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 18121*	Lemma for embedsetcestrc 18131. (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 18122*	Lemma 4 for funcsetcestrc 18128. (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 18123*	Lemma 5 for funcsetcestrc 18128. (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 18124*	Lemma 6 for funcsetcestrc 18128. (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 18125*	Lemma 7 for funcsetcestrc 18128. (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 18126*	Lemma 8 for funcsetcestrc 18128. (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 18127*	Lemma 9 for funcsetcestrc 18128. (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 18128*	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 18129*	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 18130*	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 18131*	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 18132	Extend class notation with the product of two categories.
class ×c
Syntax	c1stf 18133	Extend class notation with the first projection functor.
class 1stF
Syntax	c2ndf 18134	Extend class notation with the second projection functor.
class 2ndF
Syntax	cprf 18135	Extend class notation with the functor pairing operation.
class ⟨,⟩F
Definition	df-xpc 18136*	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 18137*	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 18138*	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 18139*	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 18140	The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ ×c Fn (V × V)
Theorem	xpcval 18141*	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 18142	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 18143*	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 18144	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 18145	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 18146*	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 18147	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 18148	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 18149	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 18150	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 18151	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 18152*	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 18153	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 18154	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 18155*	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 18156	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 18157	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 18158*	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 18159	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 18160	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 18161	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 18162	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 18163*	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 18164	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 18165*	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 18166	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 18167	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 18168	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 18169	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 18170	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 18171	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	xpcpropd 18172	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 18173	Extend class notation with the evaluation functor.
class evalF
Syntax	ccurf 18174	Extend class notation with the currying of a functor.
class curryF
Syntax	cuncf 18175	Extend class notation with the uncurrying of a functor.
class uncurryF
Syntax	cdiag 18176	Extend class notation to include the diagonal functor.
class Δfunc
Definition	df-evlf 18177*	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 18178*	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 18179*	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 18180*	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 18181*	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 18182*	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 18183	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 18184	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 18185	Lemma for evlfcl 18186. (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 18186	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 18187*	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 18188*	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 18189*	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 18190	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 18191	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 18192	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 18193*	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 18194	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)))
Theorem	curf2cl 18195	The curry functor at a morphism is a 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)    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌)))
Theorem	curfcl 18196	The curry functor of a functor 𝐹:𝐶 × 𝐷⟶𝐸 is a functor curryF (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄))
Theorem	curfpropd 18197	If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))
Theorem	uncfval 18198	Value of the uncurry functor, which is the reverse of the curry functor, taking 𝐺:𝐶⟶(𝐷⟶𝐸) to uncurryF (𝐺):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))    ⇒   ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
Theorem	uncfcl 18199	The uncurry operation takes a functor 𝐹:𝐶⟶(𝐷⟶𝐸) to a functor uncurryF (𝐹):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
Theorem	uncf1 18200	Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) = ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌))