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	xpcval 18101*	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 18102	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 18103*	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 18104	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 18105	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 18106*	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 18107	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 18108	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 18109	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 18110	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 18111	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 18112*	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 18113	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 18114	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 18115*	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 18116	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 18117	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 18118*	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 18119	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 18120	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 18121	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 18122	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 18123*	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 18124	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 18125*	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 18126	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 18127	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 18128	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 18129	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 18130	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 18131	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 18132	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 18133	Extend class notation with the evaluation functor.
class evalF
Syntax	ccurf 18134	Extend class notation with the currying of a functor.
class curryF
Syntax	cuncf 18135	Extend class notation with the uncurrying of a functor.
class uncurryF
Syntax	cdiag 18136	Extend class notation to include the diagonal functor.
class Δfunc
Definition	df-evlf 18137*	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 18138*	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 18139*	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 18140*	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 18141*	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 18142*	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 18143	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 18144	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 18145	Lemma for evlfcl 18146. (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 18146	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 18147*	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 18148*	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 18149*	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 18150	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 18151	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 18152	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 18153*	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 18154	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 18155	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 18156	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 18157	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 18158	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 18159	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 18160	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))
Theorem	uncf2 18161	Value of the uncurry functor on a morphism. (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)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑍))    &   ⊢ (𝜑 → 𝑆 ∈ (𝑌𝐽𝑊))    ⇒   ⊢ (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd ‘𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd ‘𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌), ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st ‘𝐺)‘𝑋))𝑊)‘𝑆)))
Theorem	curfuncf 18162	Cancellation of curry with uncurry. (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 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)
Theorem	uncfcurf 18163	Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
⊢ 𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    ⇒   ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)
Theorem	diagval 18164	Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))
Theorem	diagcl 18165	The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor (𝑦 ∈ 𝐷 ↦ 𝑋) is a construction that is natural in 𝑋 (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄))
Theorem	diag1cl 18166	The constant functor of 𝑋 is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))
Theorem	diag11 18167	Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)
Theorem	diag12 18168	Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍))    ⇒   ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋))
Theorem	diag2 18169	Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
Theorem	diag2cl 18170	The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    ⇒   ⊢ (𝜑 → (𝐵 × {𝐹}) ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
Theorem	curf2ndf 18171	As shown in diagval 18164, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑦), which is a constant functor of the identity functor at 𝐷. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑄 = (𝐷 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷)))
8.4.3  Hom functor
Syntax	chof 18172	Extend class notation with the Hom functor.
class HomF
Syntax	cyon 18173	Extend class notation with the Yoneda embedding.
class Yon
Definition	df-hof 18174*	Define the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat‘𝐶) × 𝐶 to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ HomF = (𝑐 ∈ Cat ↦ ⟨(Homf ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝑐)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝑐)(2nd ‘𝑦))𝑓))))⟩)
Definition	df-yon 18175	Define the Yoneda embedding, which is the currying of the (opposite) Hom functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ Yon = (𝑐 ∈ Cat ↦ (⟨𝑐, (oppCat‘𝑐)⟩ curryF (HomF‘(oppCat‘𝑐))))
Theorem	hofval 18176*	Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat‘𝐶) × 𝐶 to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓))))⟩)
Theorem	hof1fval 18177	The object part of the Hom functor is the Homf operation, which is just a functionalized version of Hom. That is, it is a two argument function, which maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶))
Theorem	hof1 18178	The object part of the Hom functor maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(1st ‘𝑀)𝑌) = (𝑋𝐻𝑌))
Theorem	hof2fval 18179*	The morphism part of the Hom functor, for morphisms ⟨𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊⟩ (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))
Theorem	hof2val 18180*	The morphism part of the Hom functor, for morphisms ⟨𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊⟩ (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊))    ⇒   ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)))
Theorem	hof2 18181	The morphism part of the Hom functor, for morphisms ⟨𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊⟩ (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹))
Theorem	hofcllem 18182	Lemma for hofcl 18183. (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐷 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑊))    &   ⊢ (𝜑 → 𝑃 ∈ (𝑆𝐻𝑍))    &   ⊢ (𝜑 → 𝑄 ∈ (𝑊𝐻𝑇))    ⇒   ⊢ (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿)))
Theorem	hofcl 18183	Closure of the Hom functor. Note that the codomain is the category SetCat‘𝑈 for any universe 𝑈 which contains each Hom-set. This corresponds to the assertion that 𝐶 be locally small (with respect to 𝑈). (Contributed by Mario Carneiro, 15-Jan-2017.)
⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐷 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))
Theorem	oppchofcl 18184	Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑀 = (HomF‘𝑂)    &   ⊢ 𝐷 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷))
Theorem	yonval 18185	Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑀 = (HomF‘𝑂)    ⇒   ⊢ (𝜑 → 𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))
Theorem	yoncl 18186	The Yoneda embedding is a functor from the category to the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
Theorem	yon1cl 18187	The Yoneda embedding at an object of 𝐶 is a presheaf on 𝐶, also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
Theorem	yon11 18188	Value of the Yoneda embedding at an object. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘((1st ‘𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))
Theorem	yon12 18189	Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑊𝐻𝑍))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))    ⇒   ⊢ (𝜑 → (((𝑍(2nd ‘((1st ‘𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍⟩ · 𝑋)𝐹))
Theorem	yon2 18190	Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑍))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑊𝐻𝑋))    ⇒   ⊢ (𝜑 → ((((𝑋(2nd ‘𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋⟩ · 𝑍)𝐺))
Theorem	hofpropd 18191	If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (HomF‘𝐶) = (HomF‘𝐷))
Theorem	yonpropd 18192	If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (Yon‘𝐶) = (Yon‘𝐷))
Theorem	oppcyon 18193	Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑌 = (Yon‘𝑂)    &   ⊢ 𝑀 = (HomF‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑌 = (⟨𝑂, 𝐶⟩ curryF 𝑀))
Theorem	oyoncl 18194	The opposite Yoneda embedding is a functor from oppCat‘𝐶 to the functor category 𝐶 → SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑌 = (Yon‘𝑂)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝑆)    ⇒   ⊢ (𝜑 → 𝑌 ∈ (𝑂 Func 𝑄))
Theorem	oyon1cl 18195	The opposite Yoneda embedding at an object of 𝐶 is a functor from 𝐶 to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑌 = (Yon‘𝑂)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝐶 Func 𝑆))
Theorem	yonedalem1 18196	Lemma for yoneda 18207. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑇 = (SetCat‘𝑉)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ 𝐻 = (HomF‘𝑄)    &   ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   ⊢ 𝐸 = (𝑂 evalF 𝑆)    &   ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
Theorem	yonedalem21 18197	Lemma for yoneda 18207. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑇 = (SetCat‘𝑉)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ 𝐻 = (HomF‘𝑄)    &   ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   ⊢ 𝐸 = (𝑂 evalF 𝑆)    &   ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Theorem	yonedalem3a 18198*	Lemma for yoneda 18207. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑇 = (SetCat‘𝑉)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ 𝐻 = (HomF‘𝑄)    &   ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   ⊢ 𝐸 = (𝑂 evalF 𝑆)    &   ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))    ⇒   ⊢ (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎‘𝑋)‘( 1 ‘𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st ‘𝑍)𝑋)⟶(𝐹(1st ‘𝐸)𝑋)))
Theorem	yonedalem4a 18199*	Lemma for yoneda 18207. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑇 = (SetCat‘𝑉)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ 𝐻 = (HomF‘𝑄)    &   ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   ⊢ 𝐸 = (𝑂 evalF 𝑆)    &   ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))    &   ⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋))    ⇒   ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))
Theorem	yonedalem4b 18200*	Lemma for yoneda 18207. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝑌 = (Yon‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝑇 = (SetCat‘𝑉)    &   ⊢ 𝑄 = (𝑂 FuncCat 𝑆)    &   ⊢ 𝐻 = (HomF‘𝑄)    &   ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   ⊢ 𝐸 = (𝑂 evalF 𝑆)    &   ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    &   ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)    &   ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))    &   ⊢ (𝜑 → 𝐴 ∈ ((1st ‘𝐹)‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))    ⇒   ⊢ (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd ‘𝐹)𝑃)‘𝐺)‘𝐴))