P. 175 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17401-17500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	catlid 17401	Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (( 1 ‘𝑌)(⟨𝑋, 𝑌⟩ · 𝑌)𝐹) = 𝐹)
Theorem	catrid 17402	Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑋⟩ · 𝑌)( 1 ‘𝑋)) = 𝐹)
Theorem	catcocl 17403	Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
Theorem	catass 17404	Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))    ⇒   ⊢ (𝜑 → ((𝐾(⟨𝑌, 𝑍⟩ · 𝑊)𝐺)(⟨𝑋, 𝑌⟩ · 𝑊)𝐹) = (𝐾(⟨𝑋, 𝑍⟩ · 𝑊)(𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)))
Theorem	catcone0 17405	Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅)    &   ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅)
Theorem	0catg 17406	Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
Theorem	0cat 17407	The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ ∅ ∈ Cat
Theorem	homffval 17408*	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
⊢ 𝐹 = (Homf ‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
Theorem	fnhomeqhomf 17409	If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
⊢ 𝐹 = (Homf ‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)
Theorem	homfval 17410	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐹 = (Homf ‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Theorem	homffn 17411	The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐹 = (Homf ‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐹 Fn (𝐵 × 𝐵)
Theorem	homfeq 17412*	Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    ⇒   ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
Theorem	homfeqd 17413	If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))    &   ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))    ⇒   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
Theorem	homfeqbas 17414	Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    ⇒   ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
Theorem	homfeqval 17415	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
Theorem	comfffval 17416*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Theorem	comffval 17417*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
Theorem	comfval 17418	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
Theorem	comfffval2 17419*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
Theorem	comffval2 17420*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
Theorem	comfval2 17421	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
Theorem	comfffn 17422	The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)
Theorem	comffn 17423	The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (compf‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
Theorem	comfeq 17424*	Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    ⇒   ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)))
Theorem	comfeqd 17425	Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
Theorem	comfeqval 17426	Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
Theorem	catpropd 17427	Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
Theorem	cidpropd 17428	Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))
8.1.2  Opposite category
Syntax	coppc 17429	The opposite category operation.
class oppCat
Definition	df-oppc 17430*	Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑓)(1st ‘𝑢)))⟩))
Theorem	oppcval 17431*	Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
Theorem	oppchomfval 17432	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ tpos 𝐻 = (Hom ‘𝑂)
Theorem	oppchomfvalOLD 17433	Obsolete proof of oppchomfval 17432 as of 14-Oct-2024. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ tpos 𝐻 = (Hom ‘𝑂)
Theorem	oppchom 17434	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋)
Theorem	oppccofval 17435	Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
Theorem	oppcco 17436	Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍)𝐹) = (𝐹(⟨𝑍, 𝑌⟩ · 𝑋)𝐺))
Theorem	oppcbas 17437	Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppcbasOLD 17438	Obsolete version of oppcbas 17437 as of 18-Oct-2024. Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppccatid 17439	Lemma for oppccat 17442. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))
Theorem	oppchomf 17440	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐶)    ⇒   ⊢ tpos 𝐻 = (Homf ‘𝑂)
Theorem	oppcid 17441	Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Id‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵)
Theorem	oppccat 17442	An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
Theorem	2oppcbas 17443	The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17458. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ 𝐵 = (Base‘(oppCat‘𝑂))
Theorem	2oppchomf 17444	The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17458. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
Theorem	2oppccomf 17445	The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17458. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
Theorem	oppchomfpropd 17446	If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    ⇒   ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
Theorem	oppccomfpropd 17447	If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    ⇒   ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))
Theorem	oppccatf 17448	oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ (oppCat ↾ Cat):Cat⟶Cat
8.1.3  Monomorphisms and epimorphisms
Syntax	cmon 17449	Extend class notation with the class of all monomorphisms.
class Mono
Syntax	cepi 17450	Extend class notation with the class of all epimorphisms.
class Epi
Definition	df-mon 17451*	Function returning the monomorphisms of the category 𝑐. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
Definition	df-epi 17452	Function returning the epimorphisms of the category 𝑐. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
Theorem	monfval 17453*	Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
Theorem	ismon 17454*	Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))))
Theorem	ismon2 17455*	Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ))))
Theorem	monhom 17456	A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌))
Theorem	moni 17457	Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))    ⇒   ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) ↔ 𝐺 = 𝐾))
Theorem	monpropd 17458	If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))
Theorem	oppcmon 17459	A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝑀 = (Mono‘𝑂)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Theorem	oppcepi 17460	An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐸 = (Epi‘𝑂)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋))
Theorem	isepi 17461*	Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑧)𝐹)))))
Theorem	isepi2 17462*	Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑌𝐻𝑧)∀ℎ ∈ (𝑌𝐻𝑧)((𝑔(⟨𝑋, 𝑌⟩ · 𝑧)𝐹) = (ℎ(⟨𝑋, 𝑌⟩ · 𝑧)𝐹) → 𝑔 = ℎ))))
Theorem	epihom 17463	An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋𝐻𝑌))
Theorem	epii 17464	Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐸𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → ((𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐾(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ↔ 𝐺 = 𝐾))
8.1.4  Sections, inverses, isomorphisms
Syntax	csect 17465	Extend class notation with the sections of a morphism.
class Sect
Syntax	cinv 17466	Extend class notation with the inverses of a morphism.
class Inv
Syntax	ciso 17467	Extend class notation with the class of all isomorphisms.
class Iso
Definition	df-sect 17468*	Function returning the section relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑓Sect𝑔, that is, 𝑓 is a section of 𝑔, if 𝑔 ∘ 𝑓 = 1‘𝑋. If there there is an arrow 𝑔 with 𝑓Sect𝑔, the arrow 𝑓 is called a section, see definition 7.19 of [Adamek] p. 106. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
Definition	df-inv 17469*	The inverse relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑔Inv𝑓, that is, 𝑔 is an inverse of 𝑓, if 𝑔 is a section of 𝑓 and 𝑓 is a section of 𝑔. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥))))
Definition	df-iso 17470*	Function returning the isomorphisms of the category 𝑐. Definition 3.8 of [Adamek] p. 28, and definition in [Lang] p. 54. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
Theorem	sectffval 17471*	Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩ · 𝑥)𝑓) = ( 1 ‘𝑥))}))
Theorem	sectfval 17472*	Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝑓) = ( 1 ‘𝑋))})
Theorem	sectss 17473	The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Theorem	issect 17474	The property "𝐹 is a section of 𝐺". (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))))
Theorem	issect2 17475	Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋))    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
Theorem	sectcan 17476	If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹)    &   ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻)    ⇒   ⊢ (𝜑 → 𝐺 = 𝐻)
Theorem	sectco 17477	Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)    &   ⊢ (𝜑 → 𝐻(𝑌𝑆𝑍)𝐾)    ⇒   ⊢ (𝜑 → (𝐻(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌⟩ · 𝑋)𝐾))
Theorem	isofval 17478*	Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.)
⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Theorem	invffval 17479*	Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥))))
Theorem	invfval 17480	Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)))
Theorem	isinv 17481	Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)))
Theorem	invss 17482	The inverse relation is a relation between morphisms 𝐹:𝑋⟶𝑌 and their inverses 𝐺:𝑌⟶𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
Theorem	invsym 17483	The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐺(𝑌𝑁𝑋)𝐹))
Theorem	invsym2 17484	The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
Theorem	invfun 17485	The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → Fun (𝑋𝑁𝑌))
Theorem	isoval 17486	The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
Theorem	inviso1 17487	If 𝐺 is an inverse to 𝐹, then 𝐹 is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
Theorem	inviso2 17488	If 𝐺 is an inverse to 𝐹, then 𝐺 is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋))
Theorem	invf 17489	The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
Theorem	invf1o 17490	The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism 𝐹 ∈ (𝑋𝐼𝑌) has a unique inverse, denoted by ((Inv‘𝐶)‘𝐹). Remark 3.12 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))
Theorem	invinv 17491	The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)
Theorem	invco 17492	The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌⟩ · 𝑋)((𝑌𝑁𝑍)‘𝐺)))
Theorem	dfiso2 17493*	Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)    &   ⊢ ∗ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))
Theorem	dfiso3 17494*	Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔)))
Theorem	inveq 17495	If there are two inverses of a morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.) (Revised by AV, 3-Jul-2022.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))
Theorem	isofn 17496	The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2020.)
⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
Theorem	isohom 17497	An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
Theorem	isoco 17498	The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
Theorem	oppcsect 17499	A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝑇 = (Sect‘𝑂)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ 𝐺(𝑋𝑆𝑌)𝐹))
Theorem	oppcsect2 17500	A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝑇 = (Sect‘𝑂)    ⇒   ⊢ (𝜑 → (𝑋𝑇𝑌) = ◡(𝑋𝑆𝑌))