P. 483 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48201-48300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	catprs2 48201*	A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 48202 and catprsc2 48203 for constructions satisfying the hypothesis "catprs.1". See catprs 48200 for a more primitive version. See prsthinc 48243 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ Proset )
Theorem	catprsc 48202*	A construction of the preorder induced by a category. See catprs2 48201 for details. See also catprsc2 48203 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Theorem	catprsc2 48203*	An alternate construction of the preorder induced by a category. See catprs2 48201 for details. See also catprsc 48202 for a different construction. The two constructions are different because df-cat 17651 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Theorem	endmndlem 48204	A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 48273 for converting a monoid to a category. Lemma for bj-endmnd 36925. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀))    &   ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀))    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mnd)
21.46.12.2  Monomorphisms and epimorphisms
Theorem	idmon 48205	An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))
Theorem	idepi 48206	An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋))
21.46.12.3  Functors
Theorem	funcf2lem 48207*	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))))
21.46.13  Examples of categories
21.46.13.1  Thin categories
Syntax	cthinc 48208	Extend class notation with the class of thin categories.
class ThinCat
Definition	df-thinc 48209*	Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
Theorem	isthinc 48210*	The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Theorem	isthinc2 48211*	A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
Theorem	isthinc3 48212*	A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Theorem	thincc 48213	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
Theorem	thinccd 48214	A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	thincssc 48215	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat ⊆ Cat
Theorem	isthincd2lem1 48216*	Lemma for isthincd2 48227 and thincmo2 48217. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo2 48217	Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo 48218*	There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmoALT 48219*	Alternate proof for thincmo 48218. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmod 48220*	At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincn0eu 48221*	In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	thincid 48222	In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))    ⇒   ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
Theorem	thincmon 48223	In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 48285. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Theorem	thincepi 48224	In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 48286. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Theorem	isthincd2lem2 48225*	Lemma for isthincd2 48227. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
Theorem	isthincd 48226*	The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	isthincd2 48227*	The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))
Theorem	oppcthin 48228	The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
Theorem	subthinc 48229	A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐷 ∈ ThinCat)
Theorem	functhinclem1 48230*	Lemma for functhinc 48234. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   ⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))
Theorem	functhinclem2 48231*	Lemma for functhinc 48234. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
Theorem	functhinclem3 48232*	Lemma for functhinc 48234. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))    &   ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))    &   ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	functhinclem4 48233*	Lemma for functhinc 48234. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝐸)    ⇒   ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘( 1 ‘𝑎)) = (𝐼‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑚 ∈ (𝑎𝐻𝑏)∀𝑛 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑛(⟨𝑎, 𝑏⟩ · 𝑐)𝑚)) = (((𝑏𝐺𝑐)‘𝑛)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩𝑂(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑚))))
Theorem	functhinc 48234*	A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 48201). (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))
Theorem	fullthinc 48235*	A functor to a thin category is full iff empty hom-sets are mapped to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
Theorem	fullthinc2 48236	A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
Theorem	thincfth 48237	A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
Theorem	thincciso 48238*	Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ 𝐻 = (Hom ‘𝑋)    &   ⊢ 𝐽 = (Hom ‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑌 ∈ ThinCat)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
Theorem	0thincg 48239	Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)
Theorem	0thinc 48240	The empty category (see 0cat 17672) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ∅ ∈ ThinCat
Theorem	indthinc 48241*	An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 48243, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	indthincALT 48242*	An alternate proof for indthinc 48241 assuming more axioms including ax-pow 5365 and ax-un 7741. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	prsthinc 48243*	Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 48200 and catprs2 48201 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	setcthin 48244*	A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (SetCat‘𝑈))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	setc2othin 48245	The category (SetCat‘2o) is thin. A special case of setcthin 48244. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (SetCat‘2o) ∈ ThinCat
Theorem	thincsect 48246	In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))
Theorem	thincsect2 48247	In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹))
Theorem	thincinv 48248	In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺))
Theorem	thinciso 48249	In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))
Theorem	thinccic 48250	In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅)))
21.46.13.2  Preordered sets as thin categories
Syntax	cprstc 48251	Class function defining preordered sets as categories.
class ProsetToCat
Definition	df-prstc 48252	Definition of the function converting a preordered set to a category. Justified by prsthinc 48243. This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 48255, prstchom 48266, and prstcthin 48265. Other important properties include prstcbas 48256, prstcleval 48257, prstcle 48259, prstcocval 48260, prstcoc 48262, prstchom2 48267, and prstcprs 48264. Use those instead. Note that the defining property prstchom 48266 is equivalent to prstchom2 48267 given prstcthin 48265. See thincn0eu 48221 for justification. "ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcval 48253	Lemma for prstcnidlem 48254 and prstcthin 48265. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcnidlem 48254	Lemma for prstcnid 48255 and prstchomval 48263. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
Theorem	prstcnid 48255	Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))
Theorem	prstcbas 48256	The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	prstcleval 48257	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → ≤ = (le‘𝐶))
Theorem	prstclevalOLD 48258	Obsolete proof of prstcleval 48257 as of 12-Nov-2024. Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → ≤ = (le‘𝐶))
Theorem	prstcle 48259	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌))
Theorem	prstcocval 48260	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ⊥ = (oc‘𝐶))
Theorem	prstcocvalOLD 48261	Obsolete proof of prstcocval 48260 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ⊥ = (oc‘𝐶))
Theorem	prstcoc 48262	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋))
Theorem	prstchomval 48263	Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    ⇒   ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
Theorem	prstcprs 48264	The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ Proset )
Theorem	prstcthin 48265	The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	prstchom 48266	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Theorem	prstchom2 48267*	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 48268). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	prstchom2ALT 48268*	Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 48252. See prstchom2 48267 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	postcpos 48269	The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postcposALT 48270	Alternate proof for postcpos 48269. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postc 48271*	The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦)))
21.46.13.3  Monoids as categories
Syntax	cmndtc 48272	Class function defining monoids as categories.
class MndToCat
Definition	df-mndtc 48273	Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24. The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 48275), instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 48276, mndtchom 48279, mndtcco 48280. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
Theorem	mndtcval 48274	Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
Theorem	mndtcbasval 48275	The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑀})
Theorem	mndtcbas 48276*	The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
Theorem	mndtcob 48277	Lemma for mndtchom 48279 and mndtcco 48280. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑀)
Theorem	mndtcbas2 48278	Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	mndtchom 48279	The only hom-set of the category built from a monoid is the base set 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 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀))
Theorem	mndtcco 48280	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+g‘𝑀))
Theorem	mndtcco2 48281	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍))    ⇒   ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹))
Theorem	mndtccatid 48282*	Lemma for mndtccat 48283 and mndtcid 48284. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))
Theorem	mndtccat 48283	The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	mndtcid 48284	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	grptcmon 48285	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 48286	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))
21.47  Mathbox for Emmett Weisz
21.47.1  Miscellaneous Theorems Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs.
Theorem	nfintd 48287	Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴)
Theorem	nfiund 48288*	Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2365. See nfiundg 48289 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 48289	Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2365, see nfiund 48288 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 48290*	The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7782, but does not use it directly, since ssorduni 7782 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 48291*	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 48292	Specialization deduction, using implicit substitution. Based on the proof of spimed 2381. (Contributed by Emmett Weisz, 17-Jan-2020.)
⊢ (𝜒 → Ⅎ𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓))
Theorem	spcdvw 48293*	A version of spcdv 3578 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 48294*	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 48295*	Deduction form of bnd2 9918. (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 48296*	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)))
21.47.2  Set Recursion
21.47.2.1  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 48298. The class 𝑌 is explained in the comment of setrec1lem1 48301. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs.
Syntax	csetrecs 48297	Extend class notation to include a set defined by transfinite recursion.
class setrecs(𝐹)
Definition	df-setrecs 48298*	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 48305 and setrec2v 48310, 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 48322 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 48305 and setrec2v 48310 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 48305 and setrec2v 48310), 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 8431, 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 9617! 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 5633, 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 5633; 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 9617. 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 5633! 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 5633. 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 5633. Thus, in our case, df-fr 5633 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 48299	Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Theorem	nfsetrecs 48300	Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥setrecs(𝐹)