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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | thincn0eu 46801* | In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) β β’ (π β ((ππ»π) β β β β!π π β (ππ»π))) | ||
Theorem | thincid 46802 | In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ 1 = (IdβπΆ) & β’ (π β πΉ β (ππ»π)) β β’ (π β πΉ = ( 1 βπ)) | ||
Theorem | thincmon 46803 | In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 46865. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = (MonoβπΆ) β β’ (π β (πππ) = (ππ»π)) | ||
Theorem | thincepi 46804 | In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 46866. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ πΈ = (EpiβπΆ) β β’ (π β (ππΈπ) = (ππ»π)) | ||
Theorem | isthincd2lem2 46805* | Lemma for isthincd2 46807. (Contributed by Zhi Wang, 17-Sep-2024.) |
β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) & β’ (π β βπ₯ β π΅ βπ¦ β π΅ βπ§ β π΅ βπ β (π₯π»π¦)βπ β (π¦π»π§)(π(β¨π₯, π¦β© Β· π§)π) β (π₯π»π§)) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) β (ππ»π)) | ||
Theorem | isthincd 46806* | The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β β*π π β (π₯π»π¦)) & β’ (π β πΆ β Cat) β β’ (π β πΆ β ThinCat) | ||
Theorem | isthincd2 46807* | The predicate "πΆ is a thin category" without knowing πΆ is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β β*π π β (π₯π»π¦)) & β’ (π β Β· = (compβπΆ)) & β’ (π β πΆ β π) & β’ (π β ((π₯ β π΅ β§ π¦ β π΅ β§ π§ β π΅) β§ (π β (π₯π»π¦) β§ π β (π¦π»π§)))) & β’ ((π β§ π¦ β π΅) β 1 β (π¦π»π¦)) & β’ ((π β§ π) β (π(β¨π₯, π¦β© Β· π§)π) β (π₯π»π§)) β β’ (π β (πΆ β ThinCat β§ (IdβπΆ) = (π¦ β π΅ β¦ 1 ))) | ||
Theorem | oppcthin 46808 | The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.) |
β’ π = (oppCatβπΆ) β β’ (πΆ β ThinCat β π β ThinCat) | ||
Theorem | subthinc 46809 | A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.) |
β’ π· = (πΆ βΎcat π½) & β’ (π β π½ β (SubcatβπΆ)) & β’ (π β πΆ β ThinCat) β β’ (π β π· β ThinCat) | ||
Theorem | functhinclem1 46810* | Lemma for functhinc 46814. 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.) |
β’ π΅ = (Baseβπ·) & β’ πΆ = (BaseβπΈ) & β’ π» = (Hom βπ·) & β’ π½ = (Hom βπΈ) & β’ (π β πΈ β ThinCat) & β’ (π β πΉ:π΅βΆπΆ) & β’ πΎ = (π₯ β π΅, π¦ β π΅ β¦ ((π₯π»π¦) Γ ((πΉβπ₯)π½(πΉβπ¦)))) & β’ ((π β§ (π§ β π΅ β§ π€ β π΅)) β (((πΉβπ§)π½(πΉβπ€)) = β β (π§π»π€) = β )) β β’ (π β ((πΊ β V β§ πΊ Fn (π΅ Γ π΅) β§ βπ§ β π΅ βπ€ β π΅ (π§πΊπ€):(π§π»π€)βΆ((πΉβπ§)π½(πΉβπ€))) β πΊ = πΎ)) | ||
Theorem | functhinclem2 46811* | Lemma for functhinc 46814. (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β βπ₯ β π΅ βπ¦ β π΅ (((πΉβπ₯)π½(πΉβπ¦)) = β β (π₯π»π¦) = β )) β β’ (π β (((πΉβπ)π½(πΉβπ)) = β β (ππ»π) = β )) | ||
Theorem | functhinclem3 46812* | Lemma for functhinc 46814. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β (ππ»π)) & β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ((π₯π»π¦) Γ ((πΉβπ₯)π½(πΉβπ¦))))) & β’ (π β (((πΉβπ)π½(πΉβπ)) = β β (ππ»π) = β )) & β’ (π β β*π π β ((πΉβπ)π½(πΉβπ))) β β’ (π β ((ππΊπ)βπ) β ((πΉβπ)π½(πΉβπ))) | ||
Theorem | functhinclem4 46813* | Lemma for functhinc 46814. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ π΅ = (Baseβπ·) & β’ πΆ = (BaseβπΈ) & β’ π» = (Hom βπ·) & β’ π½ = (Hom βπΈ) & β’ (π β π· β Cat) & β’ (π β πΈ β ThinCat) & β’ (π β πΉ:π΅βΆπΆ) & β’ πΎ = (π₯ β π΅, π¦ β π΅ β¦ ((π₯π»π¦) Γ ((πΉβπ₯)π½(πΉβπ¦)))) & β’ (π β βπ§ β π΅ βπ€ β π΅ (((πΉβπ§)π½(πΉβπ€)) = β β (π§π»π€) = β )) & β’ 1 = (Idβπ·) & β’ πΌ = (IdβπΈ) & β’ Β· = (compβπ·) & β’ π = (compβπΈ) β β’ ((π β§ πΊ = πΎ) β βπ β π΅ (((ππΊπ)β( 1 βπ)) = (πΌβ(πΉβπ)) β§ βπ β π΅ βπ β π΅ βπ β (ππ»π)βπ β (ππ»π)((ππΊπ)β(π(β¨π, πβ© Β· π)π)) = (((ππΊπ)βπ)(β¨(πΉβπ), (πΉβπ)β©π(πΉβπ))((ππΊπ)βπ)))) | ||
Theorem | functhinc 46814* | 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 46781). (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ π΅ = (Baseβπ·) & β’ πΆ = (BaseβπΈ) & β’ π» = (Hom βπ·) & β’ π½ = (Hom βπΈ) & β’ (π β π· β Cat) & β’ (π β πΈ β ThinCat) & β’ (π β πΉ:π΅βΆπΆ) & β’ πΎ = (π₯ β π΅, π¦ β π΅ β¦ ((π₯π»π¦) Γ ((πΉβπ₯)π½(πΉβπ¦)))) & β’ (π β βπ§ β π΅ βπ€ β π΅ (((πΉβπ§)π½(πΉβπ€)) = β β (π§π»π€) = β )) β β’ (π β (πΉ(π· Func πΈ)πΊ β πΊ = πΎ)) | ||
Theorem | fullthinc 46815* | 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.) |
β’ π΅ = (BaseβπΆ) & β’ π½ = (Hom βπ·) & β’ π» = (Hom βπΆ) & β’ (π β π· β ThinCat) & β’ (π β πΉ(πΆ Func π·)πΊ) β β’ (π β (πΉ(πΆ Full π·)πΊ β βπ₯ β π΅ βπ¦ β π΅ ((π₯π»π¦) = β β ((πΉβπ₯)π½(πΉβπ¦)) = β ))) | ||
Theorem | fullthinc2 46816 | A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ π΅ = (BaseβπΆ) & β’ π½ = (Hom βπ·) & β’ π» = (Hom βπΆ) & β’ (π β π· β ThinCat) & β’ (π β πΉ(πΆ Full π·)πΊ) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β ((ππ»π) = β β ((πΉβπ)π½(πΉβπ)) = β )) | ||
Theorem | thincfth 46817 | A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.) |
β’ (π β πΆ β ThinCat) & β’ (π β πΉ(πΆ Func π·)πΊ) β β’ (π β πΉ(πΆ Faith π·)πΊ) | ||
Theorem | thincciso 46818* | 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.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (Baseβπ) & β’ π = (Baseβπ) & β’ π» = (Hom βπ) & β’ π½ = (Hom βπ) & β’ (π β π β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β ThinCat) & β’ (π β π β ThinCat) β β’ (π β (π( βπ βπΆ)π β βπ(βπ₯ β π βπ¦ β π ((π₯π»π¦) = β β ((πβπ₯)π½(πβπ¦)) = β ) β§ π:π β1-1-ontoβπ))) | ||
Theorem | 0thincg 46819 | Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.) |
β’ ((πΆ β π β§ β = (BaseβπΆ)) β πΆ β ThinCat) | ||
Theorem | 0thinc 46820 | The empty category (see 0cat 17503) is thin. (Contributed by Zhi Wang, 17-Sep-2024.) |
β’ β β ThinCat | ||
Theorem | indthinc 46821* | 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 β . This is a special case of prsthinc 46823, where β€ = (π΅ Γ π΅). 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.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β ((π΅ Γ π΅) Γ {1o}) = (Hom βπΆ)) & β’ (π β β = (compβπΆ)) & β’ (π β πΆ β π) β β’ (π β (πΆ β ThinCat β§ (IdβπΆ) = (π¦ β π΅ β¦ β ))) | ||
Theorem | indthincALT 46822* | An alternate proof for indthinc 46821 assuming more axioms including ax-pow 5318 and ax-un 7662. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β ((π΅ Γ π΅) Γ {1o}) = (Hom βπΆ)) & β’ (π β β = (compβπΆ)) & β’ (π β πΆ β π) β β’ (π β (πΆ β ThinCat β§ (IdβπΆ) = (π¦ β π΅ β¦ β ))) | ||
Theorem | prsthinc 46823* | 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 46780 and catprs2 46781 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.) |
β’ (π β π΅ = (BaseβπΆ)) & β’ (π β ( β€ Γ {1o}) = (Hom βπΆ)) & β’ (π β β = (compβπΆ)) & β’ (π β β€ = (leβπΆ)) & β’ (π β πΆ β Proset ) β β’ (π β (πΆ β ThinCat β§ (IdβπΆ) = (π¦ β π΅ β¦ β ))) | ||
Theorem | setcthin 46824* | A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (SetCatβπ)) & β’ (π β π β π) & β’ (π β βπ₯ β π β*π π β π₯) β β’ (π β πΆ β ThinCat) | ||
Theorem | setc2othin 46825 | The category (SetCatβ2o) is thin. A special case of setcthin 46824. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (SetCatβ2o) β ThinCat | ||
Theorem | thincsect 46826 | 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.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = (SectβπΆ) & β’ π» = (Hom βπΆ) β β’ (π β (πΉ(πππ)πΊ β (πΉ β (ππ»π) β§ πΊ β (ππ»π)))) | ||
Theorem | thincsect2 46827 | In a thin category, πΉ is a section of πΊ iff πΊ is a section of πΉ. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = (SectβπΆ) β β’ (π β (πΉ(πππ)πΊ β πΊ(πππ)πΉ)) | ||
Theorem | thincinv 46828 | In a thin category, πΉ is an inverse of πΊ iff πΉ is a section of πΊ (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π = (SectβπΆ) & β’ π = (InvβπΆ) β β’ (π β (πΉ(πππ)πΊ β πΉ(πππ)πΊ)) | ||
Theorem | thinciso 46829 | In a thin category, πΉ:πβΆπ is an isomorphism iff there is a morphism from π to π. (Contributed by Zhi Wang, 25-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π» = (Hom βπΆ) & β’ πΌ = (IsoβπΆ) & β’ (π β πΉ β (ππ»π)) β β’ (π β (πΉ β (ππΌπ) β (ππ»π) β β )) | ||
Theorem | thinccic 46830 | In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.) |
β’ (π β πΆ β ThinCat) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ π» = (Hom βπΆ) β β’ (π β (π( βπ βπΆ)π β ((ππ»π) β β β§ (ππ»π) β β ))) | ||
Syntax | cprstc 46831 | Class function defining preordered sets as categories. |
class ProsetToCat | ||
Definition | df-prstc 46832 |
Definition of the function converting a preordered set to a category.
Justified by prsthinc 46823.
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 46835, prstchom 46846, and prstcthin 46845. Other important properties include prstcbas 46836, prstcleval 46837, prstcle 46839, prstcocval 46840, prstcoc 46842, prstchom2 46847, and prstcprs 46844. Use those instead. Note that the defining property prstchom 46846 is equivalent to prstchom2 46847 given prstcthin 46845. See thincn0eu 46801 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.) |
β’ ProsetToCat = (π β Proset β¦ ((π sSet β¨(Hom βndx), ((leβπ) Γ {1o})β©) sSet β¨(compβndx), β β©)) | ||
Theorem | prstcval 46833 | Lemma for prstcnidlem 46834 and prstcthin 46845. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ = ((πΎ sSet β¨(Hom βndx), ((leβπΎ) Γ {1o})β©) sSet β¨(compβndx), β β©)) | ||
Theorem | prstcnidlem 46834 | Lemma for prstcnid 46835 and prstchomval 46843. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (compβndx) β β’ (π β (πΈβπΆ) = (πΈβ(πΎ sSet β¨(Hom βndx), ((leβπΎ) Γ {1o})β©))) | ||
Theorem | prstcnid 46835 | Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (compβndx) & β’ (πΈβndx) β (Hom βndx) β β’ (π β (πΈβπΎ) = (πΈβπΆ)) | ||
Theorem | prstcbas 46836 | The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β π΅ = (BaseβπΎ)) β β’ (π β π΅ = (BaseβπΆ)) | ||
Theorem | prstcleval 46837 | 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β β€ = (leβπΆ)) | ||
Theorem | prstclevalOLD 46838 | Obsolete proof of prstcleval 46837 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β β€ = (leβπΆ)) | ||
Theorem | prstcle 46839 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΎ)) β β’ (π β (π β€ π β π(leβπΆ)π)) | ||
Theorem | prstcocval 46840 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β β₯ = (ocβπΆ)) | ||
Theorem | prstcocvalOLD 46841 | Obsolete proof of prstcocval 46840 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β β₯ = (ocβπΆ)) | ||
Theorem | prstcoc 46842 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β₯ = (ocβπΎ)) β β’ (π β ( β₯ βπ) = ((ocβπΆ)βπ)) | ||
Theorem | prstchomval 46843 | Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) β β’ (π β ( β€ Γ {1o}) = (Hom βπΆ)) | ||
Theorem | prstcprs 46844 | The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ β Proset ) | ||
Theorem | prstcthin 46845 | The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β πΆ β ThinCat) | ||
Theorem | prstchom 46846 |
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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π β (BaseβπΆ)) & β’ (π β π β (BaseβπΆ)) β β’ (π β (π β€ π β (ππ»π) β β )) | ||
Theorem | prstchom2 46847* |
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 46848). 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π β (BaseβπΆ)) & β’ (π β π β (BaseβπΆ)) β β’ (π β (π β€ π β β!π π β (ππ»π))) | ||
Theorem | prstchom2ALT 46848* | Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 46832. See prstchom2 46847 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.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ (π β β€ = (leβπΆ)) & β’ (π β π» = (Hom βπΆ)) β β’ (π β (π β€ π β β!π π β (ππ»π))) | ||
Theorem | postcpos 46849 | The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β (πΎ β Poset β πΆ β Poset)) | ||
Theorem | postcposALT 46850 | Alternate proof for postcpos 46849. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) β β’ (π β (πΎ β Poset β πΆ β Poset)) | ||
Theorem | postc 46851* | The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.) |
β’ (π β πΆ = (ProsetToCatβπΎ)) & β’ (π β πΎ β Proset ) & β’ π΅ = (BaseβπΆ) β β’ (π β (πΆ β Poset β βπ₯ β π΅ βπ¦ β π΅ (π₯( βπ βπΆ)π¦ β π₯ = π¦))) | ||
Syntax | cmndtc 46852 | Class function defining monoids as categories. |
class MndToCat | ||
Definition | df-mndtc 46853 |
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 46855) , 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 46856, mndtchom 46859, mndtcco 46860. 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.) |
β’ MndToCat = (π β Mnd β¦ {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©}) | ||
Theorem | mndtcval 46854 | Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β πΆ = {β¨(Baseβndx), {π}β©, β¨(Hom βndx), {β¨π, π, (Baseβπ)β©}β©, β¨(compβndx), {β¨β¨π, π, πβ©, (+gβπ)β©}β©}) | ||
Theorem | mndtcbasval 46855 | The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) β β’ (π β π΅ = {π}) | ||
Theorem | mndtcbas 46856* | The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) β β’ (π β β!π₯ π₯ β π΅) | ||
Theorem | mndtcob 46857 | Lemma for mndtchom 46859 and mndtcco 46860. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) β β’ (π β π = π) | ||
Theorem | mndtcbas2 46858 | Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β π = π) | ||
Theorem | mndtchom 46859 | 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.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) β β’ (π β (ππ»π) = (Baseβπ)) | ||
Theorem | mndtcco 46860 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β Β· = (compβπΆ)) β β’ (π β (β¨π, πβ© Β· π) = (+gβπ)) | ||
Theorem | mndtcco2 46861 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β Β· = (compβπΆ)) & β’ (π β β¬ = (β¨π, πβ© Β· π)) β β’ (π β (πΊ β¬ πΉ) = (πΊ(+gβπ)πΉ)) | ||
Theorem | mndtccatid 46862* | Lemma for mndtccat 46863 and mndtcid 46864. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β (πΆ β Cat β§ (IdβπΆ) = (π¦ β (BaseβπΆ) β¦ (0gβπ)))) | ||
Theorem | mndtccat 46863 | The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) β β’ (π β πΆ β Cat) | ||
Theorem | mndtcid 46864 | 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.) |
β’ (π β πΆ = (MndToCatβπ)) & β’ (π β π β Mnd) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β 1 = (IdβπΆ)) β β’ (π β ( 1 βπ) = (0gβπ)) | ||
Theorem | grptcmon 46865 | All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπΊ)) & β’ (π β πΊ β Grp) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) & β’ (π β π = (MonoβπΆ)) β β’ (π β (πππ) = (ππ»π)) | ||
Theorem | grptcepi 46866 | All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
β’ (π β πΆ = (MndToCatβπΊ)) & β’ (π β πΊ β Grp) & β’ (π β π΅ = (BaseβπΆ)) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π» = (Hom βπΆ)) & β’ (π β πΈ = (EpiβπΆ)) β β’ (π β (ππΈπ) = (ππ»π)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 46867 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
β’ (π β β²π₯π΄) β β’ (π β β²π₯β© π΄) | ||
Theorem | nfiund 46868* | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2371. See nfiundg 46869 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
β’ β²π₯π & β’ (π β β²π¦π΄) & β’ (π β β²π¦π΅) β β’ (π β β²π¦βͺ π₯ β π΄ π΅) | ||
Theorem | nfiundg 46869 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2371, see nfiund 46868 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
β’ β²π₯π & β’ (π β β²π¦π΄) & β’ (π β β²π¦π΅) β β’ (π β β²π¦βͺ π₯ β π΄ π΅) | ||
Theorem | iunord 46870* | The indexed union of a collection of ordinal numbers π΅(π₯) is ordinal. This proof is based on the proof of ssorduni 7703, but does not use it directly, since ssorduni 7703 does not work when π΅ is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
β’ (βπ₯ β π΄ Ord π΅ β Ord βͺ π₯ β π΄ π΅) | ||
Theorem | iunordi 46871* | The indexed union of a collection of ordinal numbers π΅(π₯) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
β’ Ord π΅ β β’ Ord βͺ π₯ β π΄ π΅ | ||
Theorem | spd 46872 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2387. (Contributed by Emmett Weisz, 17-Jan-2020.) |
β’ (π β β²π₯π) & β’ (π₯ = π¦ β (π β π)) β β’ (π β (βπ₯π β π)) | ||
Theorem | spcdvw 46873* | A version of spcdv 3551 where π and π are direct substitutions of each other. This theorem is useful because it does not require π and π₯ to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
β’ (π β π΄ β π΅) & β’ (π₯ = π΄ β (π β π)) β β’ (π β (βπ₯π β π)) | ||
Theorem | tfis2d 46874* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
β’ (π β (π₯ = π¦ β (π β π))) & β’ (π β (π₯ β On β (βπ¦ β π₯ π β π))) β β’ (π β (π₯ β On β π)) | ||
Theorem | bnd2d 46875* | Deduction form of bnd2 9762. (Contributed by Emmett Weisz, 19-Jan-2021.) |
β’ (π β π΄ β V) & β’ (π β βπ₯ β π΄ βπ¦ β π΅ π) β β’ (π β βπ§(π§ β π΅ β§ βπ₯ β π΄ βπ¦ β π§ π)) | ||
Theorem | dffun3f 46876* | Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.) |
β’ β²π₯π΄ & β’ β²π¦π΄ & β’ β²π§π΄ β β’ (Fun π΄ β (Rel π΄ β§ βπ₯βπ§βπ¦(π₯π΄π¦ β π¦ = π§))) | ||
Symbols in this section: All the symbols used in the definition of setrecs(πΉ) are explained in the comment of df-setrecs 46878. The class π is explained in the comment of setrec1lem1 46881. Glossaries of symbols used in individual proofs, or used differently in different proofs, are in the comments of those proofs. | ||
Syntax | csetrecs 46877 | Extend class notation to include a set defined by transfinite recursion. |
class setrecs(πΉ) | ||
Definition | df-setrecs 46878* |
Define a class setrecs(πΉ) by transfinite recursion, where
(πΉβπ₯) is the set of new elements to add to
the class given the
set π₯ 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(πΉ). The goal of this definition is to
construct a class
fulfilling Theorems setrec1 46885 and setrec2v 46890, which give a more
intuitive idea of the meaning of setrecs.
Unlike wrecs,
setrecs is well-defined for any πΉ and
meaningful for any
function πΉ.
For example, see Theorem onsetrec 46901 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 π§, and consider the result when applying πΉ to any subset π€ β π§. Remember that πΉ 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 π§. However, if we restrict the domain of πΉ to a given set π¦, the resulting range will be a set. Therefore, with this restricted πΉ, it makes sense to consider sets π§ that are closed under πΉ applied to its subsets. Now we can test whether a given set π¦ is recursively generated by πΉ. If every set π§ that is closed under πΉ contains π¦, that means that every member of π¦ must eventually be generated by πΉ. On the other hand, if some such π§ does not contain a certain element of π¦, then that element can be avoided even if we apply πΉ in every possible way to previously generated elements. Note that such an omitted element might be eventually recursively generated by πΉ, but not through the elements of π¦. In this case, π¦ would fail the condition in the definition, but the omitted element would still be included in some larger π¦. For example, if πΉ is the successor function, the set {β , 2o} would fail the condition since 2o is not an element of the successor of β or {β }. Remember that we are applying πΉ to subsets of π¦, not elements of π¦. In fact, even the set {1o} fails the condition, since the only subset of previously generated elements is β , and suc β does not have 1o as an element. However, we can let π¦ be any ordinal, since each of its elements is generated by starting with β and repeatedly applying the successor function. A similar definition I initially used for setrecs(πΉ) was setrecs(πΉ) = βͺ ran recs((π β V β¦ (πΉββͺ ran π))). I had initially tried and failed to find an elementary definition, and I had proven theorems analogous to setrec1 46885 and setrec2v 46890 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(πΉ) is that the necessary properties to prove are self-referential (see setrec1 46885 and setrec2v 46890), 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 8323, 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(πΉ) as a union or intersection of suitable sets. Thus, instead of using a class abstraction for the elements of setrecs(πΉ), which seems to be impossible, we can use a class abstraction for supersets or subsets of setrecs(πΉ), which "know" about multiple individual elements at a time. Note that we cannot define setrecs(πΉ) 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(πΉ) as a union of a class abstraction. If setrecs(πΉ) = βͺ π΄, the elements of A must be subsets of setrecs(πΉ) which together include everything recursively generated by πΉ. We can do this by letting π΄ be the class of sets π₯ whose elements are all recursively generated by πΉ. One necessary condition is that each element of a given π₯ β π΄ must be generated by πΉ when applied to a previous element π¦ β π΄. In symbols, βπ₯ β π΄βπ¦ β π΄(π¦ β π₯ β§ π₯ β (πΉβπ¦))}. However, this is not sufficient. All fixed points π₯ of πΉ will satisfy this condition whether they should be in setrecs(πΉ) or not. If we replace the subset relation with the proper subset relation, π₯ cannot be the empty set, even though the empty set should be in π΄. 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 π¦ β π΄ to be generated by πΉ when applied only to sets of previously generated elements within π¦. For example, taking πΉ to be the successor function, we can let π΄ = On rather than π« On, and we will still have βͺ π΄ = On as required. This gets rid of the circularity of the definition, since we should have a condition to test whether a given set π¦ is in π΄ without knowing about any of the other elements of π΄. The definition I ended up using accomplishes this using induction: π΄ is defined as the class of sets π¦ for which a sort of induction on the elements of π¦ 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 π = {β¨π, πβ© β£ (πΉβπ) β π}, then (π Fr π΄ β βπ₯((π₯ β π΄ β§ π₯ β β ) β βπ¦ β π₯βπ§ β π₯Β¬ (πΉβπ§) β π¦)). On 22-Jul-2020 I came up with the following definition (Version 1) phrased in terms of induction: βͺ {π¦ β£ βπ§ (βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} 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 π§ replaced by its complement. βͺ {π¦ β£ βπ§ (π¦ β π§ β βπ€(π€ β π¦ β§ (πΉβπ€) β π§ β§ Β¬ π€ β π§))} These definitions didn't work because the induction didn't "get off the ground." If π§ does not contain the empty set, the condition (βπ€...π¦ β π§ fails, so π¦ = β doesn't get included in π΄ 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 π§ of all individuals that have been generated so far. So one approach is to replace every occurrence of β π§ with β π§, making π§ 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 πΉ be the power set function, then the induction in the inductive version works for π§ being the class of transitive sets, restricted to subsets of π¦. Therefore, π¦ must be transitive by definition of π§. This doesn't affect the union of all such π¦, but it may or may not be desirable. The problem is that πΉ is only applied to transitive sets, because of the strong requirement π€ β π§, so the definition requires the additional constraint (π β π β (πΉβπ) β (πΉβπ)) in order to work. This issue can also be avoided by replacing β π§ with β π§. The induction version of the result is used in the final definition. Version 2: (18-Aug-2020) Induction: βͺ {π¦ β£ βπ§ (βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§(π¦ β© π§ β β β βπ€(π€ β π¦ β§ π€ β© π§ = β β§ (πΉβπ€) β© π§ β β ))} In the induction version, not only does π§ include all the elements of π¦, but it must include the elements of (πΉβπ€) for π€ β (π¦ β© π§) even if those elements of (πΉβπ€) are not in π¦. We shouldn't care about any of the elements of π§ outside π¦, but this detail doesn't affect the correctness of the definition. If we replaced (πΉβπ€) in the definition by ((πΉβπ€) β© π¦), we would get the same class for setrecs(πΉ). Suppose we could find a π§ for which the condition fails for a given π¦ under the changed definition. Then the antecedent would be true, but π¦ β π§ would be false. We could then simply add all elements of (πΉβπ€) outside of π¦ for any π€ β π¦, 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 π§ fails the condition under the Metamath definition. The other direction is easier. If a certain π§ fails the Metamath definition, then all (πΉβπ€) β π§ for π€ β (π¦ β© π§), and in particular ((πΉβπ€) β© π¦) β π§. The foundedness version is starting to look more like ax-reg 9461! 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 π§ which are subsets of π¦, we can restrict π§ to π¦ in the foundedness definition. Furthermore, instead of quantifying over π€, quantify over the elements π£ β π§ overlapping with π€. Versions 3, 4, and 5 are all equivalent to Version 2. Version 3 - Foundedness (5-Sep-2020): βͺ {π¦ β£ βπ§((π§ β π¦ β§ π§ β β ) β βπ£ β π§βπ€(π€ β π¦ β§ π€ β© π§ = β β§ π£ β (πΉβπ€)))} Now, if we replace (πΉβπ€) by ((πΉβπ€) β© π¦), we do not change the definition. We already know that π£ β π¦ since π£ β π§ and π§ β π¦. 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 5585, we add another variable π’ representing the nonexistent element of π€ in π§. Version 4 - Foundedness (6-Sep-2020): βͺ {π¦ β£ βπ§((π§ β π¦ β§ π§ β β ) β βπ£ β π§βπ€βπ’ β π§(π€ β π¦ β§ Β¬ π’ β π€ β§ π£ β (πΉβπ€)) This is so close to df-fr 5585; the only change needed is to switch βπ€ with βπ’ β π§. 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 9461. Rather than a disjoint element of π§, there's a disjoint coverer of an element of π§. Finally, here's a different dead end I followed: To clean up our foundedness definition, we keep π§ as a family of sets π¦ but allow π€ to be any subset of βͺ π§ in the induction. With this stronger induction, we can also allow for the stronger requirement π« π¦ β π§ rather than only π¦ β π§. This will help improve the foundedness version. Version 1.1 (28-Aug-2020) Induction: βͺ {π¦ β£ βπ§(βπ€ (π€ β π¦ β (π€ β βͺ π§ β (πΉβπ€) β π§)) β π« π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§(βπ(π β π¦ β§ π β π§) β βπ€(π€ β π¦ β§ π€ β© β© π§ = β β§ (πΉβπ€) β π§))} ( Edit (Aug 31) - this isn't true! Nothing forces the subset of an element of π§ to be in π§. Version 2 does not have this issue. ) Similarly, we could allow π€ to be any subset of any element of π§ rather than any subset of βͺ π§. 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 π§ which are subsets of π¦, we can restrict π§ to π« π¦ in the foundedness definition: Version 1.2 (31-Aug-2020) Foundedness: βͺ {π¦ β£ βπ§((π§ β π« π¦ β§ π§ β β ) β βπ€(π€ β π« π¦ β§ π€ β© β© π§ = β β§ (πΉβπ€) β π§))} Now this looks more like df-fr 5585! The last step necessary to be able to use Fr directly in our definition is to replace (πΉβπ€) with its own setvar variable, corresponding to π¦ in df-fr 5585. This definition is incorrect, though, since there's nothing forcing the subset of an element of π§ to be in π§. Version 1.3 (31-Aug-2020) Induction: βͺ {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β βͺ π§ β (π€ β π§ β§ (πΉβπ€) β π§))) β π« π¦ β π§)} Foundedness: βͺ {π¦ β£ βπ§((π§ β π« π¦ β§ π§ β β ) β βπ€(π€ β π« π¦ β§ π€ β© β© π§ = β β§ (π€ β π§ β¨ (πΉβπ€) β π§)))} π§ must contain the supersets of each of its elements in the foundedness version, and we can't make any restrictions on π§ or πΉ, so this doesn't work. Let's try letting R be the covering relation π = {β¨π, πβ© β£ π β (πΉβπ)} to solve the transitivity issue (i.e. that if πΉ is the power set relation, π΄ consists only of transitive sets). The set (πΉβπ€) corresponds to the variable π¦ in df-fr 5585. Thus, in our case, df-fr 5585 is equivalent to (π Fr π΄ β βπ§((π§ β π΄ β§ π§ β β ) β βπ€((πΉβπ€) β π§ β§ Β¬ βπ£ β π§π£π (πΉβπ€))). Substituting our relation π gives (π Fr π΄ β βπ§((π§ β π΄ β§ π§ β β ) β βπ€((πΉβπ€) β π§ β§ Β¬ βπ£ β π§(πΉβπ€) β (πΉβπ£))) This doesn't work for non-injective πΉ because we need all π§ 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 Β¬ π€ β π§ β βπ£ β π§Β¬ π£π (πΉβπ€) so we can replace one with the other. Negate both sides: π€ β π§ β βπ£ β π§π£π (πΉβπ€) If πΉ 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 π€ 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.) |
β’ setrecs(πΉ) = βͺ {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} | ||
Theorem | setrecseq 46879 | Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.) |
β’ (πΉ = πΊ β setrecs(πΉ) = setrecs(πΊ)) | ||
Theorem | nfsetrecs 46880 | Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
β’ β²π₯πΉ β β’ β²π₯setrecs(πΉ) | ||
Theorem | setrec1lem1 46881* |
Lemma for setrec1 46885. This is a utility theorem showing the
equivalence
of the statement π β π and its expanded form. The proof
uses
elabg 3626 and equivalence theorems.
Variable π is the class of sets π¦ that are recursively generated by the function πΉ. In other words, π¦ β π iff by starting with the empty set and repeatedly applying πΉ to subsets π€ of our set, we will eventually generate all the elements of π. In this theorem, π is any element of π, and π is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π β π) β β’ (π β (π β π β βπ§(βπ€(π€ β π β (π€ β π§ β (πΉβπ€) β π§)) β π β π§))) | ||
Theorem | setrec1lem2 46882* | Lemma for setrec1 46885. If a family of sets are all recursively generated by πΉ, so is their union. In this theorem, π is a family of sets which are all elements of π, and π is any class. Use dfss3 3930, 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.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π β π) & β’ (π β π β π) β β’ (π β βͺ π β π) | ||
Theorem | setrec1lem3 46883* | Lemma for setrec1 46885. If each element π of π΄ is covered by a set π₯ recursively generated by πΉ, then there is a single such set covering all of π΄. The set is constructed explicitly using setrec1lem2 46882. It turns out that π₯ = π΄ also works, i.e., given the hypotheses it is possible to prove that π΄ β π. I don't know if proving this fact directly using setrec1lem1 46881 would be any easier than the current proof using setrec1lem2 46882, and it would only slightly simplify the proof of setrec1 46885. Other than the use of bnd2d 46875, this is a purely technical theorem for rearranging notation from that of setrec1lem2 46882 to that of setrec1 46885. (Contributed by Emmett Weisz, 20-Jan-2021.) (New usage is discouraged.) |
β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π΄ β V) & β’ (π β βπ β π΄ βπ₯(π β π₯ β§ π₯ β π)) β β’ (π β βπ₯(π΄ β π₯ β§ π₯ β π)) | ||
Theorem | setrec1lem4 46884* |
Lemma for setrec1 46885. If π is recursively generated by πΉ, then
so is π βͺ (πΉβπ΄).
In the proof of setrec1 46885, the following is substituted for this theorem's π: (π β§ (π΄ β π₯ β§ π₯ β {π¦ β£ βπ§(βπ€ (π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)})) Therefore, we cannot declare π§ to be a distinct variable from π, since we need it to appear as a bound variable in π. This theorem can be proven without the hypothesis β²π§π, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1837, making the antecedent of each line something more complicated than π. The proof of setrec1lem2 46882 could similarly be made easier to read by adding the hypothesis β²π§π, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
β’ β²π§π & β’ π = {π¦ β£ βπ§(βπ€(π€ β π¦ β (π€ β π§ β (πΉβπ€) β π§)) β π¦ β π§)} & β’ (π β π΄ β V) & β’ (π β π΄ β π) & β’ (π β π β π) β β’ (π β (π βͺ (πΉβπ΄)) β π) | ||
Theorem | setrec1 46885 |
This is the first of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(πΉ) is closed under πΉ. This
effectively sets the
actual value of setrecs(πΉ) as a lower bound for
setrecs(πΉ), as it implies that any set
generated by successive
applications of πΉ is a member of π΅. This
theorem "gets off the
ground" because we can start by letting π΄ = β
, and the
hypotheses
of the theorem will hold trivially.
Variable π΅ represents an abbreviation of setrecs(πΉ) or another name of setrecs(πΉ) (for an example of the latter, see theorem setrecon). Proof summary: Assume that π΄ β π΅, meaning that all elements of π΄ are in some set recursively generated by πΉ. Then by setrec1lem3 46883, π΄ is a subset of some set recursively generated by πΉ. (It turns out that π΄ itself is recursively generated by πΉ, but we don't need this fact. See the comment to setrec1lem3 46883.) Therefore, by setrec1lem4 46884, (πΉβπ΄) is a subset of some set recursively generated by πΉ. Thus, by ssuni 4891, it is a subset of the union of all sets recursively generated by πΉ. See df-setrecs 46878 for a detailed description of how the setrecs definition works. (Contributed by Emmett Weisz, 9-Oct-2020.) |
β’ π΅ = setrecs(πΉ) & β’ (π β π΄ β V) & β’ (π β π΄ β π΅) β β’ (π β (πΉβπ΄) β π΅) | ||
Theorem | setrec2fun 46886* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(πΉ) is a subclass of all classes πΆ that
are closed
under πΉ. Taken together, Theorems setrec1 46885 and setrec2v 46890 say
that setrecs(πΉ) is the minimal class closed under
πΉ.
We express this by saying that if πΉ respects the β relation and πΆ is closed under πΉ, then π΅ β πΆ. By substituting strategically constructed classes for πΆ, we can easily prove many useful properties. Although this theorem cannot show equality between π΅ and πΆ, if we intend to prove equality between π΅ and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7779) to the other class. (Contributed by Emmett Weisz, 15-Feb-2021.) (New usage is discouraged.) |
β’ β²ππΉ & β’ π΅ = setrecs(πΉ) & β’ Fun πΉ & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setrec2lem1 46887* | Lemma for setrec2 46889. The functional part of πΉ has the same values as πΉ. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
β’ ((πΉ βΎ {π₯ β£ β!π¦ π₯πΉπ¦})βπ) = (πΉβπ) | ||
Theorem | setrec2lem2 46888* | Lemma for setrec2 46889. The functional part of πΉ is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.) |
β’ Fun (πΉ βΎ {π₯ β£ β!π¦ π₯πΉπ¦}) | ||
Theorem | setrec2 46889* |
This is the second of two fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(πΉ) is a subclass of all classes πΆ that
are closed
under πΉ. Taken together, Theorems setrec1 46885 and setrec2v 46890
uniquely determine setrecs(πΉ) to be the minimal class closed
under πΉ.
We express this by saying that if πΉ respects the β relation and πΆ is closed under πΉ, then π΅ β πΆ. By substituting strategically constructed classes for πΆ, we can easily prove many useful properties. Although this theorem cannot show equality between π΅ and πΆ, if we intend to prove equality between π΅ and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7779) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
β’ β²ππΉ & β’ π΅ = setrecs(πΉ) & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setrec2v 46890* | Version of setrec2 46889 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.) |
β’ π΅ = setrecs(πΉ) & β’ (π β βπ(π β πΆ β (πΉβπ) β πΆ)) β β’ (π β π΅ β πΆ) | ||
Theorem | setis 46891* | Version of setrec2 46889 expressed as an induction schema. This theorem is a generalization of tfis3 7784. (Contributed by Emmett Weisz, 27-Feb-2022.) |
β’ π΅ = setrecs(πΉ) & β’ (π = π΄ β (π β π)) & β’ (π β βπ(βπ β π π β βπ β (πΉβπ)π)) β β’ (π β (π΄ β π΅ β π)) | ||
Theorem | elsetrecslem 46892* | Lemma for elsetrecs 46893. Any element of setrecs(πΉ) is generated by some subset of setrecs(πΉ). This is much weaker than setrec2v 46890. To see why this lemma also requires setrec1 46885, consider what would happen if we replaced π΅ with {π΄}. The antecedent would still hold, but the consequent would fail in general. Consider dispensing with the deduction form. (Contributed by Emmett Weisz, 11-Jul-2021.) (New usage is discouraged.) |
β’ π΅ = setrecs(πΉ) β β’ (π΄ β π΅ β βπ₯(π₯ β π΅ β§ π΄ β (πΉβπ₯))) | ||
Theorem | elsetrecs 46893* | A set π΄ is an element of setrecs(πΉ) iff π΄ is generated by some subset of setrecs(πΉ). The proof requires both setrec1 46885 and setrec2 46889, but this theorem is not strong enough to uniquely determine setrecs(πΉ). If πΉ respects the subset relation, the theorem still holds if both occurrences of β are replaced by β for a stronger version of the theorem. (Contributed by Emmett Weisz, 12-Jul-2021.) |
β’ π΅ = setrecs(πΉ) β β’ (π΄ β π΅ β βπ₯(π₯ β π΅ β§ π΄ β (πΉβπ₯))) | ||
Theorem | setrecsss 46894 | The setrecs operator respects the subset relation between two functions πΉ and πΊ. (Contributed by Emmett Weisz, 13-Mar-2022.) |
β’ (π β Fun πΊ) & β’ (π β πΉ β πΊ) β β’ (π β setrecs(πΉ) β setrecs(πΊ)) | ||
Theorem | setrecsres 46895 | A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
β’ π΅ = setrecs(πΉ) & β’ (π β Fun πΉ) β β’ (π β π΅ = setrecs((πΉ βΎ π« π΅))) | ||
Theorem | vsetrec 46896 | Construct V using set recursion. The proof indirectly uses trcl 9597, which relies on rec, but theoretically πΆ in trcl 9597 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable π rather than π₯ to avoid a distinct variable requirement between πΉ and π₯. (Contributed by Emmett Weisz, 23-Jun-2021.) |
β’ πΉ = (π₯ β V β¦ π« π₯) β β’ setrecs(πΉ) = V | ||
Theorem | 0setrec 46897 | If a function sends the empty set to itself, the function will not recursively generate any sets, regardless of its other values. (Contributed by Emmett Weisz, 23-Jun-2021.) |
β’ (π β (πΉββ ) = β ) β β’ (π β setrecs(πΉ) = β ) | ||
Theorem | onsetreclem1 46898* | Lemma for onsetrec 46901. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (πΉβπ) = {βͺ π, suc βͺ π} | ||
Theorem | onsetreclem2 46899* | Lemma for onsetrec 46901. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (π β On β (πΉβπ) β On) | ||
Theorem | onsetreclem3 46900* | Lemma for onsetrec 46901. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
β’ πΉ = (π₯ β V β¦ {βͺ π₯, suc βͺ π₯}) β β’ (π β On β π β (πΉβπ)) |
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