P. 500 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49901-50000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	discsubc 49901*	A discrete category, whose only morphisms are the identity morphisms, is a subcategory. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
Theorem	iinfconstbaslem 49902*	Lemma for iinfconstbas 49903. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))    ⇒   ⊢ (𝜑 → 𝐽 ∈ 𝐴)
Theorem	iinfconstbas 49903*	The discrete category is the indexed intersection of all subcategories with the same base. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)}))    ⇒   ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧)))
Theorem	nelsubclem 49904*	Lemma for nelsubc 49905. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    &   ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   ⊢ 𝐻 = (Homf ‘𝐶)    ⇒   ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓))))
Theorem	nelsubc 49905*	An empty "hom-set" for non-empty base satisfies all conditions for a subcategory but the existence of identity morphisms. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    &   ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Theorem	nelsubc2 49906	An empty "hom-set" for non-empty base is not a subcategory. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    &   ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅}))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → ¬ 𝐽 ∈ (Subcat‘𝐶))
Theorem	nelsubc3lem 49907*	Lemma for nelsubc3 49908. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ 𝐶 ∈ Cat    &   ⊢ 𝐽 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ (¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))    ⇒   ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))))
Theorem	nelsubc3 49908*	Remark 4.2(2) of [Adamek] p. 48. There exists a set satisfying all conditions for a subcategory but the existence of identity morphisms. Therefore such condition in df-subc 17893 is necessary. Note that this theorem cheated a little bit because (𝐶 ↾cat 𝐽) is not a category. In fact (𝐶 ↾cat 𝐽) ∈ Cat is a stronger statement than the condition (d) of Definition 4.1(1) of [Adamek] p. 48, as stated here (see the proof of issubc3 17930). To construct such a category, see setc1onsubc 50439 and cnelsubc 50441. (Contributed by Zhi Wang, 5-Nov-2025.)
⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧))))
Theorem	ssccatid 49909*	A category 𝐶 restricted by 𝐽 is a category if all of the following are satisfied: a) the base is a subset of base of 𝐶, b) all hom-sets are subsets of hom-sets of 𝐶, c) it has identity morphisms for all objects, d) the composition under 𝐶 is closed in 𝐽. But 𝐽 might not be a subcategory of 𝐶 (see cnelsubc 50441). (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ (𝑦𝐽𝑦))    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))    ⇒   ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 )))
Theorem	resccatlem 49910*	Lemma for resccat 49911. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑆 = (Base‘𝐸)    &   ⊢ 𝐽 = (Homf ‘𝐸)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
Theorem	resccat 49911*	A class 𝐶 restricted by the hom-sets of another set 𝐸, whose base is a subset of the base of 𝐶 and whose composition is compatible with 𝐶, is a category iff 𝐸 is a category. Note that the compatibility condition "resccat.1" can be weakened by removing 𝑥 ∈ 𝑆 because 𝑓 ∈ (𝑥𝐽𝑦) implies these. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑆 = (Base‘𝐸)    &   ⊢ 𝐽 = (Homf ‘𝐸)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓))    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat))
21.51.15.7  Functors
Theorem	reldmfunc 49912	The domain of Func is a relation. (Contributed by Zhi Wang, 12-Nov-2025.)
⊢ Rel dom Func
Theorem	func1st2nd 49913	Rewrite the functor predicate with separated parts. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
Theorem	func1st 49914	Extract the first member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
Theorem	func2nd 49915	Extract the second member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
Theorem	funcrcl2 49916	Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.)
⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐷 ∈ Cat)
Theorem	funcrcl3 49917	Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.)
⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐸 ∈ Cat)
Theorem	funcf2lem 49918*	A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	funcf2lem2 49919*	A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ 𝐵 = (𝐸‘𝐶)    ⇒   ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	0funcglem 49920	Lemma for 0funcg 49922. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜂))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜁))    &   ⊢ (𝜑 → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁)))
Theorem	0funcg2 49921	The functor from the empty category. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ∅ = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹 = ∅ ∧ 𝐺 = ∅)))
Theorem	0funcg 49922	The functor from the empty category. Corollary of Definition 3.47 of [Adamek] p. 40, Definition 7.1 of [Adamek] p. 101, Example 3.3(4.c) of [Adamek] p. 24, and Example 7.2(3) of [Adamek] p. 101. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ∅ = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐶 Func 𝐷) = {⟨∅, ∅⟩})
Theorem	0funclem 49923	Lemma for 0funcALT 49925. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))    &   ⊢ (𝜒 ↔ 𝜂)    &   ⊢ (𝜃 ↔ 𝜁)    &   ⊢ 𝜏    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁)))
Theorem	0func 49924	The functor from the empty category. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof shortened by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})
Theorem	0funcALT 49925	Alternate proof of 0func 49924. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩})
Theorem	func0g 49926	The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐵 = ∅)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	func0g2 49927	The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐵 = ∅)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	initc 49928*	Sets with empty base are the only initial objects in the category of small categories. Example 7.2(3) of [Adamek] p. 101. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ ((𝐶 ∈ V ∧ ∅ = (Base‘𝐶)) ↔ ∀𝑑 ∈ Cat ∃!𝑓 𝑓 ∈ (𝐶 Func 𝑑))
Theorem	cofu1st2nd 49929	Rewrite the functor composition with separated functor parts. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))
Theorem	rescofuf 49930	The restriction of functor composition is a function from product functor space to functor space. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)
Theorem	cofu1a 49931	Value of the object part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋))
Theorem	cofu2a 49932	Value of the morphism part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅))
Theorem	cofucla 49933	The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸))
Theorem	funchomf 49934	Source categories of a functor have the same set of objects and morphisms. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)    &   ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
Theorem	idfurcl 49935	Reverse closure for an identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ ((idfunc‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat)
Theorem	idfu1stf1o 49936	The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵)
Theorem	idfu1stalem 49937	Lemma for idfu1sta 49938. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	idfu1sta 49938	Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    ⇒   ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
Theorem	idfu1a 49939	Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
Theorem	idfu2nda 49940	Value of the morphism part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐷))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (𝑋(Hom ‘𝐷)𝑌))    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ 𝐻))
Theorem	imasubclem1 49941*	Lemma for imasubc 49988. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V)
Theorem	imasubclem2 49942*	Lemma for imasubc 49988. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ 𝐾 = (𝑦 ∈ 𝑋, 𝑧 ∈ 𝑌 ↦ ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷))    ⇒   ⊢ (𝜑 → 𝐾 Fn (𝑋 × 𝑌))
Theorem	imasubclem3 49943*	Lemma for imasubc 49988. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷))    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷))
Theorem	imaf1homlem 49944	Lemma for imaf1hom 49945 and other theorems. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵))
Theorem	imaf1hom 49945*	The hom-set of an image of a functor injective on objects. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌))))
Theorem	imaidfu2lem 49946	Lemma for imaidfu2 49948. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷))
Theorem	imaidfu 49947*	The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Homf ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴)    ⇒   ⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)
Theorem	imaidfu2 49948*	The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸))    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Homf ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝑆 = (Base‘𝐷))    ⇒   ⊢ (𝜑 → 𝐽 = 𝐾)
Theorem	cofid1a 49949	Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    ⇒   ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋)
Theorem	cofid2a 49950	Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅)
Theorem	cofid1 49951	Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    ⇒   ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋)
Theorem	cofid2 49952	Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = 𝑅)
Theorem	cofidvala 49953*	The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   ⊢ 𝐻 = (Hom ‘𝐷)    ⇒   ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
Theorem	cofidf2a 49954	If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌)) ∧ (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)):(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌)))
Theorem	cofidf1a 49955	If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the object part of 𝐹 is injective, and the object part of 𝐺 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼)    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → ((1st ‘𝐹):𝐵–1-1→𝐶 ∧ (1st ‘𝐺):𝐶–onto→𝐵))
Theorem	cofidval 49956*	The property "⟨𝐹, 𝐺⟩ is a section of ⟨𝐾, 𝐿⟩ " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   ⊢ 𝐻 = (Hom ‘𝐷)    ⇒   ⊢ (𝜑 → ((𝐾 ∘ 𝐹) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))))
Theorem	cofidf2 49957	If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ∧ ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)𝐽(𝐹‘𝑌))–onto→(𝑋𝐻𝑌)))
Theorem	cofidf1 49958	If "⟨𝐹, 𝐺⟩ is a section of ⟨𝐾, 𝐿⟩ " in a category of small categories (in a universe), then 𝐹 is injective, and 𝐾 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → (𝐹:𝐵–1-1→𝐶 ∧ 𝐾:𝐶–onto→𝐵))
21.51.15.8  Opposite functors
Syntax	coppf 49959	Extend class notation with the operation generating opposite functors.
class oppFunc
Definition	df-oppf 49960*	Definition of the operation generating opposite functors. Definition 3.41 of [Adamek] p. 39. The object part of the functor is unchanged while the morphism part is transposed due to reversed direction of arrows in the opposite category. The opposite functor is a functor on opposite categories (oppfoppc 49978). (Contributed by Zhi Wang, 4-Nov-2025.) Better reverse closure. (Revised by Zhi Wang, 13-Nov-2025.)
⊢ oppFunc = (𝑓 ∈ V, 𝑔 ∈ V ↦ if((Rel 𝑔 ∧ Rel dom 𝑔), ⟨𝑓, tpos 𝑔⟩, ∅))
Theorem	oppffn 49961	oppFunc is a function on (V × V). (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ oppFunc Fn (V × V)
Theorem	reldmoppf 49962	The domain of oppFunc is a relation. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ Rel dom oppFunc
Theorem	oppfvalg 49963	Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 oppFunc 𝐺) = if((Rel 𝐺 ∧ Rel dom 𝐺), ⟨𝐹, tpos 𝐺⟩, ∅))
Theorem	oppfrcllem 49964	Lemma for oppfrcl 49965. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    ⇒   ⊢ (𝜑 → 𝐺 ≠ ∅)
Theorem	oppfrcl 49965	If an opposite functor of a class is a functor, then the original class must be an ordered pair. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (V × V))
Theorem	oppfrcl2 49966	If an opposite functor of a class is a functor, then the two components of the original class must be sets. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   ⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	oppfrcl3 49967	If an opposite functor of a class is a functor, then the second component of the original class must be a relation whose domain is a relation as well. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   ⊢ (𝜑 → (Rel 𝐵 ∧ Rel dom 𝐵))
Theorem	oppf1st2nd 49968	Rewrite the opposite functor into its components (eqopi 8028). (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩)    ⇒   ⊢ (𝜑 → (𝐺 ∈ (V × V) ∧ ((1st ‘𝐺) = 𝐴 ∧ (2nd ‘𝐺) = tpos 𝐵)))
Theorem	2oppf 49969	The double opposite functor is the original functor. Remark 3.42 of [Adamek] p. 39. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ (𝜑 → 𝐺 ∈ 𝑅)    &   ⊢ Rel 𝑅    &   ⊢ 𝐺 = ( oppFunc ‘𝐹)    ⇒   ⊢ (𝜑 → ( oppFunc ‘𝐺) = 𝐹)
Theorem	eloppf 49970	The pre-image of a non-empty opposite functor is non-empty; and the second component of the pre-image is a relation on triples. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝐺 = ( oppFunc ‘𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ≠ ∅ ∧ (Rel (2nd ‘𝐹) ∧ Rel dom (2nd ‘𝐹))))
Theorem	eloppf2 49971	Both components of a pre-image of a non-empty opposite functor exist; and the second component is a relation on triples. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝐹 oppFunc 𝐺) = 𝐾    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Rel 𝐺 ∧ Rel dom 𝐺)))
Theorem	oppfvallem 49972	Lemma for oppfval 49973. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (Rel 𝐺 ∧ Rel dom 𝐺))
Theorem	oppfval 49973	Value of the opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 oppFunc 𝐺) = ⟨𝐹, tpos 𝐺⟩)
Theorem	oppfval2 49974	Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)
⊢ (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st ‘𝐹), tpos (2nd ‘𝐹)⟩)
Theorem	oppfval3 49975	Value of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 = ⟨𝐺, 𝐾⟩)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → ( oppFunc ‘𝐹) = ⟨𝐺, tpos 𝐾⟩)
Theorem	oppf1 49976	Value of the object part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐹)) = (1st ‘𝐹))
Theorem	oppf2 49977	Value of the morphism part of the opposite functor. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝑀(2nd ‘( oppFunc ‘𝐹))𝑁) = (𝑁(2nd ‘𝐹)𝑀))
Theorem	oppfoppc 49978	The opposite functor is a functor on opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (𝐹 oppFunc 𝐺) ∈ (𝑂 Func 𝑃))
Theorem	oppfoppc2 49979	The opposite functor is a functor on opposite categories. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
Theorem	funcoppc2 49980	A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)tpos 𝐺)
Theorem	funcoppc4 49981	A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐹 oppFunc 𝐺) ∈ (𝑂 Func 𝑃))    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
Theorem	funcoppc5 49982	A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
Theorem	2oppffunc 49983	The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑃))    ⇒   ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝐶 Func 𝐷))
Theorem	funcoppc3 49984	A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)    &   ⊢ (𝜑 → 𝐺 Fn (𝐴 × 𝐵))    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
Theorem	oppff1 49985	The operation generating opposite functors is injective. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    ⇒   ⊢ ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃)
Theorem	oppff1o 49986	The operation generating opposite functors is bijective. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃))
Theorem	cofuoppf 49987	Composition of opposite functors. (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾))
21.51.15.9  Full & faithful functors
Theorem	imasubc 49988*	An image of a full functor is a full subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐽 = (Homf ‘𝐸)    ⇒   ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ 𝐶 ∧ (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾))
Theorem	imasubc2 49989*	An image of a full functor is a (full) subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))
Theorem	imassc 49990*	An image of a functor satisfies the subcategory subset relation. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ 𝐽 = (Homf ‘𝐸)    ⇒   ⊢ (𝜑 → 𝐾 ⊆cat 𝐽)
Theorem	imaid 49991*	An image of a functor preserves the identity morphism. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋))
Theorem	imaf1co 49992*	An image of a functor whose object part is injective preserves the composition. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ ∙ = (comp‘𝐸)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍))    ⇒   ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍))
Theorem	imasubc3 49993*	An image of a functor injective on objects is a subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ 𝑆 = (𝐹 “ 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝)))    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → Fun ◡𝐹)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸))
Theorem	fthcomf 49994*	Source categories of a faithful functor have the same base, hom-sets and composition operation if the composition is compatible in images of the functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺)    &   ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)    &   ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)))    ⇒   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
Theorem	idfth 49995	The inclusion functor is a faithful functor. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸))
Theorem	idemb 49996	The inclusion functor is an embedding. Remark 4.4(1) in [Adamek] p. 49. (Contributed by Zhi Wang, 16-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ◡(1st ‘𝐼)))
Theorem	idsubc 49997	The source category of an inclusion functor is a subcategory of the target category. See also Remark 4.4 in [Adamek] p. 49. (Contributed by Zhi Wang, 10-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐷)    ⇒   ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸))
Theorem	idfullsubc 49998	The source category of an inclusion functor is a full subcategory of the target category if the inclusion functor is full. Remark 4.4(2) in [Adamek] p. 49. See also ressffth 18021. (Contributed by Zhi Wang, 11-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐷)    &   ⊢ 𝐽 = (Homf ‘𝐸)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻))
Theorem	cofidfth 49999	If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49958, this theorem proves that 𝐹 is an embedding (a faithful functor injective on objects, remark 3.28(1) of [Adamek] p. 34). (Contributed by Zhi Wang, 15-Nov-2025.)
⊢ 𝐼 = (idfunc‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿)    &   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼)    ⇒   ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺)
Theorem	fulloppf 50000	The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Zhi Wang, 26-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))    ⇒   ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Full 𝑃))