P. 508 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50701-50800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grptcmon 50701	All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑀 = (Mono‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Theorem	grptcepi 50702	All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐸 = (Epi‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
21.51.16.6  Categories with at most one object and at most two morphisms
Theorem	2arwcatlem1 50703	Lemma for 2arwcat 50708. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝑋𝐻𝑋) = { 0 , 1 }    ⇒   ⊢ ((((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 ))) ↔ ((𝑥 ∈ {𝑋} ∧ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ {𝑋} ∧ 𝑤 ∈ {𝑋}) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
Theorem	2arwcatlem2 50704	Lemma for 2arwcat 50708. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    ⇒   ⊢ (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = 𝐹)
Theorem	2arwcatlem3 50705	Lemma for 2arwcat 50708. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)
Theorem	2arwcatlem4 50706	Lemma for 2arwcat 50708. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝑋)    &   ⊢ (𝜑 → 𝐵 = 𝑌)    &   ⊢ (𝜑 → 𝐶 = 𝑍)    &   ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })    &   ⊢ (𝜑 → (𝐺 = 0 ∨ 𝐺 = 1 ))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })
Theorem	2arwcatlem5 50707	Lemma for 2arwcat 50708. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → ( 1 · 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 · 1 ) = 0 )    &   ⊢ (𝜑 → ( 0 · 0 ) ∈ { 0 , 1 })    ⇒   ⊢ (𝜑 → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))
Theorem	2arwcat 50708*	The condition for a structure with at most one object and at most two morphisms being a category. "2arwcat.2" to "2arwcat.5" are also necessary conditions if 𝑋, 0, and 1 are all sets, due to catlid 17857, catrid 17858, and catcocl 17859. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ (𝜑 → {𝑋} = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝑋𝐻𝑋) = { 0 , 1 }    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )    &   ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )    &   ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))
Theorem	incat 50709*	Constructing a category with at most one object and at most two morphisms. If 𝑋 is a set then 𝐶 is the category 𝐴 in Exercise 3G of [Adamek] p. 45. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), {𝑋}⟩, ⟨(Hom ‘ndx), {⟨𝑋, 𝑋, 𝐻⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑋, 𝑋⟩, 𝑋, · ⟩}⟩}    &   ⊢ 𝐻 = {𝐹, 𝐺}    &   ⊢ · = (𝑓 ∈ 𝐻, 𝑔 ∈ 𝐻 ↦ (𝑓 ∩ 𝑔))    ⇒   ⊢ ((𝐹 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 𝐺)))
Theorem	setc1onsubc 50710*	Construct a category with one object and two morphisms and prove that category (SetCat‘1o) satisfies all conditions for a subcategory but the compatibility of identity morphisms, showing the necessity of the latter condition in defining a subcategory. Exercise 4A of [Adamek] p. 58. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐶 = {⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), {⟨∅, ∅, 2o⟩}⟩, ⟨(comp‘ndx), {⟨⟨∅, ∅⟩, ∅, · ⟩}⟩}    &   ⊢ · = (𝑓 ∈ 2o, 𝑔 ∈ 2o ↦ (𝑓 ∩ 𝑔))    &   ⊢ 𝐸 = (SetCat‘1o)    &   ⊢ 𝐽 = (Homf ‘𝐸)    &   ⊢ 𝑆 = 1o    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝐷 = (𝐶 ↾cat 𝐽)    ⇒   ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ ¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat))
Theorem	cnelsubclem 50711*	Lemma for cnelsubc 50712. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐽 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ (𝐶 ∈ Cat ∧ 𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶 ↾cat 𝐽) ∈ Cat))    ⇒   ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
Theorem	cnelsubc 50712*	Remark 4.2(2) of [Adamek] p. 48. There exists a category satisfying all conditions for a subcategory but the compatibility of identity morphisms. Therefore such condition in df-subc 17987 is necessary. A stronger statement than nelsubc3 50179. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ ¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ (𝑐 ↾cat 𝑗) ∈ Cat))
21.51.17  Kan extensions and related concepts
21.51.17.1  Kan extensions
Syntax	clan 50713	Class function defining the (local) left Kan extension.
class Lan
Syntax	cran 50714	Class function defining the (local) right Kan extension.
class Ran
Definition	df-lan 50715*	Definition of the (local) left Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) consists of left Kan extensions of 𝑋 along 𝐹, which are universal pairs from 𝑋 to the pre-composition functor given by 𝐹 (lanval2 50735). See also § 3 of Chapter X in p. 240 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 50735 (retrieved 3 Nov 2025). A left Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (lanrcl4 50742) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 50743) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷. The left Kan extension is a generalization of many categorical concepts such as colimit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory". This definition was chosen over the other version in the commented out section due to its better reverse closure property. See df-ran 50716 for the dual concept. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
Definition	df-ran 50716*	Definition of the (local) right Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) consists of right Kan extensions of 𝑋 along 𝐹, which are universal pairs from the pre-composition functor given by 𝐹 to 𝑋 (ranval2 50738). The definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf 50738 (retrieved 3 Nov 2025). A right Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (ranrcl4 50747) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 50748) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷. The right Kan extension is a generalization of many categorical concepts such as limit. In § 7 of Chapter X of Categories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory". This definition was chosen over the other version in the commented out section due to its better reverse closure property. See df-lan 50715 for the dual concept. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
Theorem	lanfn 50717	Lan is a function on ((V × V) × V). (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Lan Fn ((V × V) × V)
Theorem	ranfn 50718	Ran is a function on ((V × V) × V). (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Ran Fn ((V × V) × V)
Theorem	reldmlan 50719	The domain of Lan is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom Lan
Theorem	reldmran 50720	The domain of Ran is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom Ran
Theorem	lanfval 50721*	Value of the function generating the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Lan 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ ((⟨𝐷, 𝐸⟩ −∘F 𝑓)(𝑅 UP 𝑆)𝑥)))
Theorem	ranfval 50722*	Value of the function generating the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ 𝑂 = (oppCat‘𝑅)    &   ⊢ 𝑃 = (oppCat‘𝑆)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ Ran 𝐸) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ (𝐶 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐷, 𝐸⟩ −∘F 𝑓))(𝑂 UP 𝑃)𝑥)))
Theorem	lanpropd 50723	If the categories have the same set of objects, morphisms, and compositions, then they have the same left Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   ⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))
Theorem	ranpropd 50724	If the categories have the same set of objects, morphisms, and compositions, then they have the same right Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))    &   ⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))
Theorem	reldmlan2 50725	The domain of (𝑃 Lan 𝐸) is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom (𝑃 Lan 𝐸)
Theorem	reldmran2 50726	The domain of (𝑃 Ran 𝐸) is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom (𝑃 Ran 𝐸)
Theorem	lanval 50727	Value of the set of left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝐾)    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	ranval 50728	Value of the set of right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ 𝑂 = (oppCat‘𝑅)    &   ⊢ 𝑃 = (oppCat‘𝑆)    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	lanrcl 50729	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
Theorem	ranrcl 50730	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
Theorem	rellan 50731	The set of left Kan extensions is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)
Theorem	relran 50732	The set of right Kan extensions is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)
Theorem	islan 50733	A left Kan extension is a universal pair. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) → 𝐿 ∈ (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	islan2 50734	A left Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 → 𝐿(𝐾(𝑅 UP 𝑆)𝑋)𝐴)
Theorem	lanval2 50735	The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 50302 and upciclem1 50274. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = (𝐾(𝑅 UP 𝑆)𝑋))
Theorem	isran 50736	A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋))    ⇒   ⊢ (𝜑 → 𝐿 ∈ (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	isran2 50737	A right Kan extension is a universal pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐿(⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋)𝐴)
Theorem	ranval2 50738	The set of right Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by oppcup2 50316 and oppcup3lem 50314. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐽, 𝐾⟩)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨𝐽, tpos 𝐾⟩(𝑂 UP 𝑃)𝑋))
Theorem	ranval3 50739	The set of right Kan extensions is the set of universal pairs. (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ 𝑂 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑃 = (oppCat‘(𝐶 FuncCat 𝐸))    &   ⊢ 𝐾 = (⟨𝐷, 𝐸⟩ −∘F 𝐹)    ⇒   ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (( oppFunc ‘𝐾)(𝑂 UP 𝑃)𝑋))
Theorem	lanrcl2 50740	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
Theorem	lanrcl3 50741	Reverse closure for left Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
Theorem	lanrcl4 50742	The first component of a left Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
Theorem	lanrcl5 50743	The second component of a left Kan extension is a natural transformation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴)    &   ⊢ 𝑁 = (𝐶 Nat 𝐸)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
Theorem	ranrcl2 50744	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
Theorem	ranrcl3 50745	Reverse closure for right Kan extensions. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
Theorem	ranrcl4lem 50746	Lemma for ranrcl4 50747 and ranrcl5 50748. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
Theorem	ranrcl4 50747	The first component of a right Kan extension is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
Theorem	ranrcl5 50748	The second component of a right Kan extension is a natural transformation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴)    &   ⊢ 𝑁 = (𝐶 Nat 𝐸)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
Theorem	lanup 50749*	The universal property of the left Kan extension; expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝑀 = (𝐷 Nat 𝐸)    &   ⊢ 𝑁 = (𝐶 Nat 𝐸)    &   ⊢ ∙ = (comp‘𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))    ⇒   ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))
Theorem	ranup 50750*	The universal property of the right Kan extension; expressed explicitly. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝑀 = (𝐷 Nat 𝐸)    &   ⊢ 𝑁 = (𝐶 Nat 𝐸)    &   ⊢ ∙ = (comp‘𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))    ⇒   ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹)))))
21.51.17.2  Limits and colimits
Syntax	clmd 50751	Class function defining the limit of a diagram.
class Limit
Syntax	ccmd 50752	Class function defining the colimit of a diagram.
class Colimit
Definition	df-lmd 50753*	A diagram of type 𝐷 or a 𝐷-shaped diagram in a category 𝐶, is a functor 𝐹:𝐷⟶𝐶 where the source category 𝐷, usually small or even finite, is called the index category or the scheme of the diagram. The actual objects and morphisms in 𝐷 are largely irrelevant; only the way in which they are interrelated matters. The diagram is thought of as indexing a collection of objects and morphisms in 𝐶 patterned on 𝐷. Definition 11.1(1) of [Adamek] p. 193. A cone to a diagram, or a natural source for a diagram in a category 𝐶 is a pair of an object 𝑋 in 𝐶 and a natural transformation from the constant functor (or constant diagram) of the object 𝑋 to the diagram. The second component associates each object in the index category with a morphism in 𝐶 whose domain is 𝑋 (concl 50769). The naturality guarantees that the combination of the diagram with the cone must commute (concom 50771). Definition 11.3(1) of [Adamek] p. 193. A limit of a diagram 𝐹:𝐷⟶𝐶 of type 𝐷 in category 𝐶 is a universal pair from the diagonal functor (𝐶Δfunc𝐷) to the diagram. The universal pair is a cone to the diagram satisfying the universal property, that each cone to the diagram uniquely factors through the limit (islmd 50773). Definition 11.3(2) of [Adamek] p. 194. Terminal objects (termolmd 50778), products, equalizers, pullbacks, and inverse limits can be considered as limits of some diagram; limits can be further generalized as right Kan extensions (lmdran 50779). "lmd" is short for "limit of a diagram". See df-cmd 50754 for the dual concept (lmddu 50775, cmddu 50776). (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
Definition	df-cmd 50754*	A co-cone (or cocone) to a diagram (see df-lmd 50753 for definition), or a natural sink for a diagram in a category 𝐶 is a pair of an object 𝑋 in 𝐶 and a natural transformation from the diagram to the constant functor (or constant diagram) of the object 𝑋. The second component associates each object in the index category with a morphism in 𝐶 whose codomain is 𝑋 (coccl 50770). The naturality guarantees that the combination of the diagram with the co-cone must commute (coccom 50772). Definition 11.27(1) of [Adamek] p. 202. A colimit of a diagram 𝐹:𝐷⟶𝐶 of type 𝐷 in category 𝐶 is a universal pair from the diagram to the diagonal functor (𝐶Δfunc𝐷). The universal pair is a co-cone to the diagram satisfying the universal property, that each co-cone to the diagram uniquely factors through the colimit. (iscmd 50774). Definition 11.27(2) of [Adamek] p. 202. Initial objects (initocmd 50777), coproducts, coequalizers, pushouts, and direct limits can be considered as colimits of some diagram; colimits can be further generalized as left Kan extensions (cmdlan 50780). "cmd" is short for "colimit of a diagram". See df-lmd 50753 for the dual concept (lmddu 50775, cmddu 50776). (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
Theorem	reldmlmd 50755	The domain of Limit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Rel dom Limit
Theorem	reldmcmd 50756	The domain of Colimit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Rel dom Colimit
Theorem	lmdfval 50757*	Function value of Limit. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))
Theorem	cmdfval 50758*	Function value of Colimit. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))
Theorem	lmdrcl 50759	Reverse closure for a limit of a diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝑋 ∈ ((𝐶 Limit 𝐷)‘𝐹) → 𝐹 ∈ (𝐷 Func 𝐶))
Theorem	cmdrcl 50760	Reverse closure for a colimit of a diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝑋 ∈ ((𝐶 Colimit 𝐷)‘𝐹) → 𝐹 ∈ (𝐷 Func 𝐶))
Theorem	reldmlmd2 50761	The domain of (𝐶 Limit 𝐷) is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ Rel dom (𝐶 Limit 𝐷)
Theorem	reldmcmd2 50762	The domain of (𝐶 Colimit 𝐷) is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ Rel dom (𝐶 Colimit 𝐷)
Theorem	lmdfval2 50763	The set of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
Theorem	cmdfval2 50764	The set of colimits of a diagram. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
Theorem	lmdpropd 50765	If the categories have the same set of objects, morphisms, and compositions, then they have the same limits. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (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 Limit 𝐶) = (𝐵 Limit 𝐷))
Theorem	cmdpropd 50766	If the categories have the same set of objects, morphisms, and compositions, then they have the same colimits. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (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 Colimit 𝐶) = (𝐵 Colimit 𝐷))
Theorem	rellmd 50767	The set of limits of a diagram is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ Rel ((𝐶 Limit 𝐷)‘𝐹)
Theorem	relcmd 50768	The set of colimits of a diagram is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ Rel ((𝐶 Colimit 𝐷)‘𝐹)
Theorem	concl 50769	A natural transformation from a constant functor of an object maps to morphisms whose domain is the object. Therefore, the range of the second component of a cone are morphisms with a common domain. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))    ⇒   ⊢ (𝜑 → (𝑅‘𝑌) ∈ (𝑋𝐻((1st ‘𝐹)‘𝑌)))
Theorem	coccl 50770	A natural transformation to a constant functor of an object maps to morphisms whose codomain is the object. Therefore, the range of the second component of a co-cone are morphisms with a common codomain. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))    ⇒   ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1st ‘𝐹)‘𝑌)𝐻𝑋))
Theorem	concom 50771	A cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))    ⇒   ⊢ (𝜑 → (𝑅‘𝑍) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
Theorem	coccom 50772	A co-cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))    ⇒   ⊢ (𝜑 → (𝑅‘𝑌) = ((𝑅‘𝑍)(⟨((1st ‘𝐹)‘𝑌), ((1st ‘𝐹)‘𝑍)⟩ · 𝑋)((𝑌(2nd ‘𝐹)𝑍)‘𝑀)))
Theorem	islmd 50773*	The universal property of limits of a diagram. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (((1st ‘𝐿)‘𝑋)𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (((1st ‘𝐿)‘𝑥)𝑁𝐹)∃!𝑚 ∈ (𝑥𝐻𝑋)𝑎 = (𝑗 ∈ 𝐵 ↦ ((𝑅‘𝑗)(⟨𝑥, 𝑋⟩ · ((1st ‘𝐹)‘𝑗))𝑚))))
Theorem	iscmd 50774*	The universal property of colimits of a diagram. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
Theorem	lmddu 50775	The duality of limits and colimits: limits of a diagram are colimits of an opposite diagram in opposite categories. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))
Theorem	cmddu 50776	The duality of limits and colimits: colimits of a diagram are limits of an opposite diagram in opposite categories. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝐶 Colimit 𝐷)‘𝐹) = ((𝑂 Limit 𝑃)‘𝐺))
Theorem	initocmd 50777	Initial objects are the object part of colimits of the empty diagram. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (InitO‘𝐶) = dom (∅(𝐶 Colimit ∅)∅)
Theorem	termolmd 50778	Terminal objects are the object part of limits of the empty diagram. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (TermO‘𝐶) = dom (∅(𝐶 Limit ∅)∅)
Theorem	lmdran 50779	To each limit of a diagram there is a corresponding right Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 50642). (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))    &   ⊢ 𝐿 = (𝐶Δfunc 1 )    &   ⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))    ⇒   ⊢ (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))
Theorem	cmdlan 50780	To each colimit of a diagram there is a corresponding left Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 50642). (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))    &   ⊢ 𝐿 = (𝐶Δfunc 1 )    &   ⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))    ⇒   ⊢ (𝜑 → (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Lan 𝐶)𝐹)𝑀))
21.52  Mathbox for Emmett Weisz
21.52.1  Miscellaneous Theorems Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.
Theorem	nfintd 50781	Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)
Theorem	nfiund 50782*	Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2401. See nfiundg 50783 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝐴)    &   ⊢ (𝜑 → Ⅎ𝑦𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	nfiundg 50783	Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2401, see nfiund 50782 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝐴)    &   ⊢ (𝜑 → Ⅎ𝑦𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iunord 50784*	The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7793, but does not use it directly, since ssorduni 7793 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iunordi 50785*	The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.)
⊢ Ord 𝐵    ⇒   ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵
Theorem	spd 50786	Specialization deduction, using implicit substitution. Based on the proof of spimed 2417. (Contributed by Emmett Weisz, 17-Jan-2020.)
⊢ (𝜒 → Ⅎ𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓))
Theorem	tfis2d 50787*	Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.)
⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    &   ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓)))    ⇒   ⊢ (𝜑 → (𝑥 ∈ On → 𝜓))
21.52.2  Examples and properties of set recursion
Theorem	setrecseq 50788	Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Theorem	nfsetrecs 50789	Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥setrecs(𝐹)
Theorem	setrec2mpt 50790*	Version of setrec2 9977 where 𝐹 is defined using maps-to notation. Deduction form is omitted in the second hypothesis for simplicity. In practice, nothing important is lost since we are only interested in one choice of 𝐴, 𝑆, and 𝑉 at a time. However, we are interested in what happens when 𝐶 varies, so deduction form is used in the third hypothesis. (Contributed by Emmett Weisz, 4-Jun-2024.)
⊢ 𝐵 = setrecs((𝑎 ∈ 𝐴 ↦ 𝑆))    &   ⊢ (𝑎 ∈ 𝐴 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	setis 50791*	Version of setrec2 9977 expressed as an induction schema. This theorem is a generalization of tfis3 7869. (Contributed by Emmett Weisz, 27-Feb-2022.)
⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒))
Theorem	elsetrecslem 50792*	Lemma for elsetrecs 50793. Any element of setrecs(𝐹) is generated by some subset of setrecs(𝐹). This is much weaker than setrec2v 9978. To see why this lemma also requires setrec1 9972, consider what would happen if we replaced 𝐵 with {𝐴}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.)
⊢ 𝐵 = setrecs(𝐹)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
Theorem	elsetrecs 50793*	A set 𝐴 is an element of setrecs(𝐹) iff 𝐴 is generated by some subset of setrecs(𝐹). The proof requires both setrec1 9972 and setrec2 9977, but this theorem is not strong enough to uniquely determine setrecs(𝐹). If 𝐹 respects the subset relation, the theorem still holds if both occurrences of ∈ are replaced by ⊆ for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.)
⊢ 𝐵 = setrecs(𝐹)    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 ∈ (𝐹‘𝑥)))
Theorem	setrecsss 50794	The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.)
⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐺)    ⇒   ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))
Theorem	setrecsres 50795	A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.)
⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵)))
Theorem	vsetrec 50796	Construct V using set recursion. The proof indirectly uses trcl 9729, which relies on rec, but theoretically 𝐶 in trcl 9729 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.)
⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥)    ⇒   ⊢ setrecs(𝐹) = V
Theorem	0setrec 50797	If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.)
⊢ (𝜑 → (𝐹‘∅) = ∅)    ⇒   ⊢ (𝜑 → setrecs(𝐹) = ∅)
Theorem	onsetreclem1 50798*	Lemma for onsetrec 50801. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
Theorem	onsetreclem2 50799*	Lemma for onsetrec 50801. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)
Theorem	onsetreclem3 50800*	Lemma for onsetrec 50801. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})    ⇒   ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))