P. 503 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50201-50300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mndtcid 50201	The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 1 = (Id‘𝐶))    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀))
Theorem	oppgoppchom 50202	The converted opposite monoid has the same hom-set as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 21-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷))    &   ⊢ (𝜑 → 𝐽 = (Hom ‘𝑂))    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑌𝐽𝑌))
Theorem	oppgoppcco 50203	The converted opposite monoid has the same composition as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    &   ⊢ (𝜑 → · = (comp‘𝐷))    &   ⊢ (𝜑 → ∙ = (comp‘𝑂))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (⟨𝑌, 𝑌⟩ ∙ 𝑌))
Theorem	oppgoppcid 50204	The converted opposite monoid has the same identity morphism as that of the opposite category. Example 3.6(2) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐷 = (MndToCat‘(oppg‘𝑀)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑂))    ⇒   ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) = ((Id‘𝑂)‘𝑌))
Theorem	grptcmon 50205	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 50206	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))
21.49.16.6  Categories with at most one object and at most two morphisms
Theorem	2arwcatlem1 50207	Lemma for 2arwcat 50212. (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 50208	Lemma for 2arwcat 50212. (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 50209	Lemma for 2arwcat 50212. (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 50210	Lemma for 2arwcat 50212. (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 50211	Lemma for 2arwcat 50212. (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 50212*	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 17725, catrid 17726, and catcocl 17727. (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 50213*	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 50214*	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 50215*	Lemma for cnelsubc 50216. (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 50216*	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 17855 is necessary. A stronger statement than nelsubc3 49683. (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))
21.49.17  Kan extensions and related concepts
21.49.17.1  Kan extensions
Syntax	clan 50217	Class function defining the (local) left Kan extension.
class Lan
Syntax	cran 50218	Class function defining the (local) right Kan extension.
class Ran
Definition	df-lan 50219*	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 50239). 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 50239 (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 50246) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 50247) 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 50220 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 50220*	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 50242). 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 50242 (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 50251) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 50252) 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 50219 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 50221	Lan is a function on ((V × V) × V). (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Lan Fn ((V × V) × V)
Theorem	ranfn 50222	Ran is a function on ((V × V) × V). (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Ran Fn ((V × V) × V)
Theorem	reldmlan 50223	The domain of Lan is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom Lan
Theorem	reldmran 50224	The domain of Ran is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom Ran
Theorem	lanfval 50225*	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 50226*	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 50227	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 50228	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 50229	The domain of (𝑃 Lan 𝐸) is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel dom (𝑃 Lan 𝐸)
Theorem	reldmran2 50230	The domain of (𝑃 Ran 𝐸) is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel dom (𝑃 Ran 𝐸)
Theorem	lanval 50231	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 50232	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 50233	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 50234	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 50235	The set of left Kan extensions is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)
Theorem	relran 50236	The set of right Kan extensions is a relation. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ Rel (𝐹(𝑃 Ran 𝐸)𝑋)
Theorem	islan 50237	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 50238	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 50239	The set of left Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by isup2 49806 and upciclem1 49778. (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 50240	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 50241	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 50242	The set of right Kan extensions is the set of universal pairs. Therefore, the explicit universal property can be recovered by oppcup2 49820 and oppcup3lem 49818. (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 50243	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 50244	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 50245	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 50246	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 50247	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 50248	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 50249	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 50250	Lemma for ranrcl4 50251 and ranrcl5 50252. (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 50251	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 50252	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 50253*	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 50254*	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)))))
21.49.17.2  Limits and colimits
Syntax	clmd 50255	Class function defining the limit of a diagram.
class Limit
Syntax	ccmd 50256	Class function defining the colimit of a diagram.
class Colimit
Definition	df-lmd 50257*	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 50273). The naturality guarantees that the combination of the diagram with the cone must commute (concom 50275). 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 50277). Definition 11.3(2) of [Adamek] p. 194. Terminal objects (termolmd 50282), products, equalizers, pullbacks, and inverse limits can be considered as limits of some diagram; limits can be further generalized as right Kan extensions (lmdran 50283). "lmd" is short for "limit of a diagram". See df-cmd 50258 for the dual concept (lmddu 50279, cmddu 50280). (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 50258*	A co-cone (or cocone) to a diagram (see df-lmd 50257 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 50274). The naturality guarantees that the combination of the diagram with the co-cone must commute (coccom 50276). 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 50278). Definition 11.27(2) of [Adamek] p. 202. Initial objects (initocmd 50281), coproducts, coequalizers, pushouts, and direct limits can be considered as colimits of some diagram; colimits can be further generalized as left Kan extensions (cmdlan 50284). "cmd" is short for "colimit of a diagram". See df-lmd 50257 for the dual concept (lmddu 50279, cmddu 50280). (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 50259	The domain of Limit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Rel dom Limit
Theorem	reldmcmd 50260	The domain of Colimit is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Rel dom Colimit
Theorem	lmdfval 50261*	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 50262*	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 50263	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 50264	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 50265	The domain of (𝐶 Limit 𝐷) is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ Rel dom (𝐶 Limit 𝐷)
Theorem	reldmcmd2 50266	The domain of (𝐶 Colimit 𝐷) is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ Rel dom (𝐶 Colimit 𝐷)
Theorem	lmdfval2 50267	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 50268	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 50269	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 50270	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 50271	The set of limits of a diagram is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ Rel ((𝐶 Limit 𝐷)‘𝐹)
Theorem	relcmd 50272	The set of colimits of a diagram is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ Rel ((𝐶 Colimit 𝐷)‘𝐹)
Theorem	concl 50273	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 50274	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 50275	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 50276	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 50277*	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 50278*	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 50279	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 50280	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 50281	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 50282	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 50283	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 50146). (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 50284	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 50146). (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))
21.50  Mathbox for Emmett Weisz
21.50.1  Miscellaneous Theorems Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.
Theorem	nfintd 50285	Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)
Theorem	nfiund 50286*	Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2404. See nfiundg 50287 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 50287	Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2404, see nfiund 50286 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 50288*	The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7762, but does not use it directly, since ssorduni 7762 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 50289*	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 50290	Specialization deduction, using implicit substitution. Based on the proof of spimed 2420. (Contributed by Emmett Weisz, 17-Jan-2020.)
⊢ (𝜒 → Ⅎ𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓))
Theorem	spcdvw 50291*	A version of spcdv 3554 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	tfis2d 50292*	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))
Theorem	bnd2d 50293*	Deduction form of bnd2 9863. (Contributed by Emmett Weisz, 19-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓))
Theorem	dffun3f 50294*	Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑧𝐴    ⇒   ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
21.50.2  Set Recursion
21.50.2.1  Basic Properties of Set Recursion Symbols in this section: All the symbols used in the definition of setrecs(𝐹) are explained in the comment of df-setrecs 50296. The class 𝑌 is explained in the comment of setrec1lem1 50299. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.
Syntax	csetrecs 50295	Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
Definition	df-setrecs 50296*	Define a class setrecs(𝐹) by transfinite recursion, where (𝐹‘𝑥) is the set of new elements to add to the class given the set 𝑥 of elements in the class so far. We do not need a base case, because we can start with the empty set, which is vacuously a subset of setrecs(𝐹). The goal of this definition is to construct a class fulfilling Theorems setrec1 50303 and setrec2v 50308, which give a more intuitive idea of the meaning of setrecs. Unlike wrecs, setrecs is well-defined for any 𝐹 and meaningful for any function 𝐹. For example, see Theorem onsetrec 50320 for how the class On is defined recursively using the successor function. The definition works by building subsets of the desired class and taking the union of those subsets. To find such a collection of subsets, consider an arbitrary set 𝑧, and consider the result when applying 𝐹 to any subset 𝑤 ⊆ 𝑧. Remember that 𝐹 can be any function, and in general we are interested in functions that give outputs that are larger than their inputs, so we have no reason to expect the outputs to be within 𝑧. However, if we restrict the domain of 𝐹 to a given set 𝑦, the resulting range will be a set. Therefore, with this restricted 𝐹, it makes sense to consider sets 𝑧 that are closed under 𝐹 applied to its subsets. Now we can test whether a given set 𝑦 is recursively generated by 𝐹. If every set 𝑧 that is closed under 𝐹 contains 𝑦, that means that every member of 𝑦 must eventually be generated by 𝐹. On the other hand, if some such 𝑧 does not contain a certain element of 𝑦, then that element can be avoided even if we apply 𝐹 in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by 𝐹, but not through the elements of 𝑦. In this case, 𝑦 would fail the condition in the definition, but the omitted element would still be included in some larger 𝑦. For example, if 𝐹 is the successor function, the set {∅, 2o} would fail the condition since 2o is not an element of the successor of ∅ or {∅}. Remember that we are applying 𝐹 to subsets of 𝑦, not elements of 𝑦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is ∅, and suc ∅ does not have 1o as an element. However, we can let 𝑦 be any ordinal, since each of its elements is generated by starting with ∅ and repeatedly applying the successor function. A similar definition I initially used for setrecs(𝐹) was setrecs(𝐹) = ∪ ran recs((𝑔 ∈ V ↦ (𝐹‘∪ ran 𝑔))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 50303 and setrec2v 50308 using the old definition before I found the new one. I decided to change definitions for two reasons. First, as John Horton Conway noted in the Appendix to Part Zero of On Numbers and Games, mathematicians should not be caught up in any particular formalization, such as ZF set theory. Instead, they should work under whatever framework best suits the problem, and the formal bases used for different problems can be shown to be equivalent. Thus, Conway preferred defining surreal numbers as equivalence classes of surreal number forms, rather than sign-expansions. Although sign-expansions are easier to implement in ZF set theory, Conway argued that "formalisation within some particular axiomatic set theory is irrelevant". Furthermore, one of the most remarkable properties of the theory of surreal numbers is that it generates so much from almost nothing. Using sign-expansions as the formal definition destroys the beauty of surreal numbers, because ordinals are already built in. For this reason, I replaced the old definition of setrecs, which also relied heavily on ordinal numbers. On the other hand, both surreal numbers and the elementary definition of setrecs immediately generate the ordinal numbers from a (relatively) very simple set-theoretical basis. Second, although it is still complicated to formalize the theory of recursively generated sets within ZF set theory, it is actually simpler and more natural to do so with set theory directly than with the theory of ordinal numbers. As Conway wrote, indexing the "birthdays" of sets is and should be unnecessary. Using an elementary definition for setrecs removes the reliance on the previously developed theory of ordinal numbers, allowing proofs to be simpler and more direct. Formalizing surreal numbers within Metamath is probably still not in the spirit of Conway. He said that "attempts to force arbitrary theories into a single formal straitjacket... produce unnecessarily cumbrous and inelegant contortions." Nevertheless, Metamath has proven to be much more versatile than it seems at first, and I think the theory of surreal numbers can be natural while fitting well into the Metamath framework. The difficulty in writing a definition in Metamath for setrecs(𝐹) is that the necessary properties to prove are self-referential (see setrec1 50303 and setrec2v 50308), so we cannot simply write the properties we want inside a class abstraction as with most definitions. As noted in the comment of df-rdg 8381, this is not actually a requirement of the Metamath language, but we would like to be able to eliminate all definitions by direct mechanical substitution. We cannot define setrecs using a class abstraction directly, because nothing about its individual elements tells us whether they are in the set. We need to know about previous elements first. One way of getting around this problem without indexing is by defining setrecs(𝐹) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(𝐹), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(𝐹), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(𝐹) as an intersection of sets, because in general it is a proper class, so any supersets would also be proper classes. However, a proper class can be a union of sets, as long as the collection of such sets is a proper class. Therefore, it is feasible to define setrecs(𝐹) as a union of a class abstraction. If setrecs(𝐹) = ∪ 𝐴, the elements of A must be subsets of setrecs(𝐹) which together include everything recursively generated by 𝐹. We can do this by letting 𝐴 be the class of sets 𝑥 whose elements are all recursively generated by 𝐹. One necessary condition is that each element of a given 𝑥 ∈ 𝐴 must be generated by 𝐹 when applied to a previous element 𝑦 ∈ 𝐴. In symbols, ∀𝑥 ∈ 𝐴∃𝑦 ∈ 𝐴(𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐹‘𝑦))}. However, this is not sufficient. All fixed points 𝑥 of 𝐹 will satisfy this condition whether they should be in setrecs(𝐹) or not. If we replace the subset relation with the proper subset relation, 𝑥 cannot be the empty set, even though the empty set should be in 𝐴. Therefore this condition cannot be used in the definition, even if we can find a way to avoid making it circular. A better strategy is to find a necessary and sufficient condition for all the elements of a set 𝑦 ∈ 𝐴 to be generated by 𝐹 when applied only to sets of previously generated elements within 𝑦. For example, taking 𝐹 to be the successor function, we can let 𝐴 = On rather than 𝒫 On, and we will still have ∪ 𝐴 = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set 𝑦 is in 𝐴 without knowing about any of the other elements of 𝐴. The definition I ended up using accomplishes this using induction: 𝐴 is defined as the class of sets 𝑦 for which a sort of induction on the elements of 𝑦 holds. However, when creating a definition for setrecs that did not rely on ordinal numbers, I tried at first to write a definition using the well-founded relation predicate, Fr. I thought that this would be simple to do once I found a suitable definition using induction, just as the least- element principle is equivalent to induction on the positive integers. If we let 𝑅 = {⟨𝑎, 𝑏⟩ ∣ (𝐹‘𝑎) ⊆ 𝑏}, then (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥∀𝑧 ∈ 𝑥¬ (𝐹‘𝑧) ⊆ 𝑦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝑦 ∈ 𝑧)} In Aug-2020 I came up with an equivalent definition with the goal of phrasing it in terms of the relation Fr. It is the contrapositive of the previous one with 𝑧 replaced by its complement. ∪ {𝑦 ∣ ∀𝑧 (𝑦 ∈ 𝑧 → ∃𝑤(𝑤 ⊆ 𝑦 ∧ (𝐹‘𝑤) ∈ 𝑧 ∧ ¬ 𝑤 ∈ 𝑧))} These definitions didn't work because the induction didn't "get off the ground." If 𝑧 does not contain the empty set, the condition (∀𝑤...𝑦 ∈ 𝑧 fails, so 𝑦 = ∅ doesn't get included in 𝐴 even though it should. This could be fixed by adding the base case as a separate requirement, but the subtler problem would remain that rather than a set of "acceptable" sets, what we really need is a collection 𝑧 of all individuals that have been generated so far. So one approach is to replace every occurrence of ∈ 𝑧 with ⊆ 𝑧, making 𝑧 a set of individuals rather than a family of sets. That solves this problem, but it complicates the foundedness version of the definition, which looked cleaner in Version 1. There was another problem with Version 1. If we let 𝐹 be the power set function, then the induction in the inductive version works for 𝑧 being the class of transitive sets, restricted to subsets of 𝑦. Therefore, 𝑦 must be transitive by definition of 𝑧. This doesn't affect the union of all such 𝑦, but it may or may not be desirable. The problem is that 𝐹 is only applied to transitive sets, because of the strong requirement 𝑤 ∈ 𝑧, so the definition requires the additional constraint (𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) in order to work. This issue can also be avoided by replacing ∈ 𝑧 with ⊆ 𝑧. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧 (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(𝑦 ∩ 𝑧 ≠ ∅ → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∩ 𝑧 ≠ ∅))} In the induction version, not only does 𝑧 include all the elements of 𝑦, but it must include the elements of (𝐹‘𝑤) for 𝑤 ⊆ (𝑦 ∩ 𝑧) even if those elements of (𝐹‘𝑤) are not in 𝑦. We shouldn't care about any of the elements of 𝑧 outside 𝑦, but this detail doesn't affect the correctness of the definition. If we replaced (𝐹‘𝑤) in the definition by ((𝐹‘𝑤) ∩ 𝑦), we would get the same class for setrecs(𝐹). Suppose we could find a 𝑧 for which the condition fails for a given 𝑦 under the changed definition. Then the antecedent would be true, but 𝑦 ⊆ 𝑧 would be false. We could then simply add all elements of (𝐹‘𝑤) outside of 𝑦 for any 𝑤 ⊆ 𝑦, which we can do because all the classes involved are sets. This is not trivial and requires the axioms of union, power set, and replacement. However, the expanded 𝑧 fails the condition under the Metamath definition. The other direction is easier. If a certain 𝑧 fails the Metamath definition, then all (𝐹‘𝑤) ⊆ 𝑧 for 𝑤 ⊆ (𝑦 ∩ 𝑧), and in particular ((𝐹‘𝑤) ∩ 𝑦) ⊆ 𝑧. The foundedness version is starting to look more like ax-reg 9538! We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝑦 in the foundedness definition. Furthermore, instead of quantifying over 𝑤, quantify over the elements 𝑣 ∈ 𝑧 overlapping with 𝑤. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ 𝑧 = ∅ ∧ 𝑣 ∈ (𝐹‘𝑤)))} Now, if we replace (𝐹‘𝑤) by ((𝐹‘𝑤) ∩ 𝑦), we do not change the definition. We already know that 𝑣 ∈ 𝑦 since 𝑣 ∈ 𝑧 and 𝑧 ⊆ 𝑦. All we need to show in order to prove that this change leads to an equivalent definition is to find To make our definition look exactly like df-fr 5601, we add another variable 𝑢 representing the nonexistent element of 𝑤 in 𝑧. Version 4 - Foundedness (6-Sep-2020): ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑣 ∈ 𝑧∃𝑤∀𝑢 ∈ 𝑧(𝑤 ⊆ 𝑦 ∧ ¬ 𝑢 ∈ 𝑤 ∧ 𝑣 ∈ (𝐹‘𝑤)) This is so close to df-fr 5601; the only change needed is to switch ∃𝑤 with ∀𝑢 ∈ 𝑧. Unfortunately, I couldn't find any way to switch the quantifiers without interfering with the definition. Maybe there is a definition equivalent to this one that uses Fr, but I couldn't find one. Yet, we can still find a remarkable similarity between Foundedness Version 2 and ax-reg 9538. Rather than a disjoint element of 𝑧, there's a disjoint coverer of an element of 𝑧. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep 𝑧 as a family of sets 𝑦 but allow 𝑤 to be any subset of ∪ 𝑧 in the induction. With this stronger induction, we can also allow for the stronger requirement 𝒫 𝑦 ⊆ 𝑧 rather than only 𝑦 ∈ 𝑧. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝐹‘𝑤) ∈ 𝑧)) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧(∃𝑎(𝑎 ⊆ 𝑦 ∧ 𝑎 ∈ 𝑧) → ∃𝑤(𝑤 ⊆ 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of 𝑧 to be in 𝑧. Version 2 does not have this issue. ) Similarly, we could allow 𝑤 to be any subset of any element of 𝑧 rather than any subset of ∪ 𝑧. I think this has the same problem. We want to take advantage of the preexisting relation Fr, which seems closely related to our foundedness definition. Since we only care about the elements of 𝑧 which are subsets of 𝑦, we can restrict 𝑧 to 𝒫 𝑦 in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝐹‘𝑤) ∈ 𝑧))} Now this looks more like df-fr 5601! The last step necessary to be able to use Fr directly in our definition is to replace (𝐹‘𝑤) with its own setvar variable, corresponding to 𝑦 in df-fr 5601. This definition is incorrect, though, since there's nothing forcing the subset of an element of 𝑧 to be in 𝑧. Version 1.3 (31-Aug-2020) Induction: ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ ∪ 𝑧 → (𝑤 ∈ 𝑧 ∧ (𝐹‘𝑤) ∈ 𝑧))) → 𝒫 𝑦 ⊆ 𝑧)} Foundedness: ∪ {𝑦 ∣ ∀𝑧((𝑧 ⊆ 𝒫 𝑦 ∧ 𝑧 ≠ ∅) → ∃𝑤(𝑤 ∈ 𝒫 𝑦 ∧ 𝑤 ∩ ∩ 𝑧 = ∅ ∧ (𝑤 ∈ 𝑧 ∨ (𝐹‘𝑤) ∈ 𝑧)))} 𝑧 must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on 𝑧 or 𝐹, so this doesn't work. Let's try letting R be the covering relation 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑏 ∈ (𝐹‘𝑎)} to solve the transitivity issue (i.e. that if 𝐹 is the power set relation, 𝐴 consists only of transitive sets). The set (𝐹‘𝑤) corresponds to the variable 𝑦 in df-fr 5601. Thus, in our case, df-fr 5601 is equivalent to (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤))). Substituting our relation 𝑅 gives (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅) → ∃𝑤((𝐹‘𝑤) ∈ 𝑧 ∧ ¬ ∃𝑣 ∈ 𝑧(𝐹‘𝑤) ∈ (𝐹‘𝑣))) This doesn't work for non-injective 𝐹 because we need all 𝑧 to be straddlers, but we don't necessarily need all-straddlers; loops within z are fine for non-injective F. Consider the foundedness form of Version 1. We want to show ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑣 ∈ 𝑧¬ 𝑣𝑅(𝐹‘𝑤) so we can replace one with the other. Negate both sides: 𝑤 ∈ 𝑧 ↔ ∃𝑣 ∈ 𝑧𝑣𝑅(𝐹‘𝑤) If 𝐹 is injective, then we should be able to pick a suitable R, being careful about the above problem for some F (for example z = transitivity) when changing the antecedent y e. z' to z =/= (/). If we're clever, we can get rid of the injectivity requirement. The forward direction of the above equivalence always holds, but the key is that although the backwards direction doesn't hold in general, we can always find some z' where it doesn't work for 𝑤 itself. If there exists a z' where the version with the w condition fails, then there exists a z' where the version with the v condition also fails. However, Version 1 is not a correct definition, so this doesn't work either. (Contributed by Emmett Weisz, 18-Aug-2020.) (New usage is discouraged.)
⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
Theorem	setrecseq 50297	Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Theorem	nfsetrecs 50298	Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥setrecs(𝐹)
Theorem	setrec1lem1 50299*	Lemma for setrec1 50303. This is a utility theorem showing the equivalence of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof uses elabg 3636 and equivalence theorems. Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
Theorem	setrec1lem2 50300*	Lemma for setrec1 50303. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3926, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)
⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌)