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Type | Label | Description |
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Statement | ||
Theorem | homarw 18001 | A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π» = (HomaβπΆ) β β’ (ππ»π) β π΄ | ||
Theorem | arwdm 18002 | The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π΅ = (BaseβπΆ) β β’ (πΉ β π΄ β (domaβπΉ) β π΅) | ||
Theorem | arwcd 18003 | The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π΅ = (BaseβπΆ) β β’ (πΉ β π΄ β (codaβπΉ) β π΅) | ||
Theorem | dmaf 18004 | The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π΅ = (BaseβπΆ) β β’ (doma βΎ π΄):π΄βΆπ΅ | ||
Theorem | cdaf 18005 | The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π΅ = (BaseβπΆ) β β’ (coda βΎ π΄):π΄βΆπ΅ | ||
Theorem | arwhom 18006 | The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) & β’ π½ = (Hom βπΆ) β β’ (πΉ β π΄ β (2nd βπΉ) β ((domaβπΉ)π½(codaβπΉ))) | ||
Theorem | arwdmcd 18007 | Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π΄ = (ArrowβπΆ) β β’ (πΉ β π΄ β πΉ = β¨(domaβπΉ), (codaβπΉ), (2nd βπΉ)β©) | ||
Syntax | cida 18008 | Extend class notation to include identity for arrows. |
class Ida | ||
Syntax | ccoa 18009 | Extend class notation to include composition for arrows. |
class compa | ||
Definition | df-ida 18010* | Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.) |
β’ Ida = (π β Cat β¦ (π₯ β (Baseβπ) β¦ β¨π₯, π₯, ((Idβπ)βπ₯)β©)) | ||
Definition | df-coa 18011* | Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a quinary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ compa = (π β Cat β¦ (π β (Arrowβπ), π β {β β (Arrowβπ) β£ (codaββ) = (domaβπ)} β¦ β¨(domaβπ), (codaβπ), ((2nd βπ)(β¨(domaβπ), (domaβπ)β©(compβπ)(codaβπ))(2nd βπ))β©)) | ||
Theorem | idafval 18012* | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ 1 = (IdβπΆ) β β’ (π β πΌ = (π₯ β π΅ β¦ β¨π₯, π₯, ( 1 βπ₯)β©)) | ||
Theorem | idaval 18013 | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ 1 = (IdβπΆ) & β’ (π β π β π΅) β β’ (π β (πΌβπ) = β¨π, π, ( 1 βπ)β©) | ||
Theorem | ida2 18014 | Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ 1 = (IdβπΆ) & β’ (π β π β π΅) β β’ (π β (2nd β(πΌβπ)) = ( 1 βπ)) | ||
Theorem | idahom 18015 | Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) & β’ π» = (HomaβπΆ) β β’ (π β (πΌβπ) β (ππ»π)) | ||
Theorem | idadm 18016 | Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) β β’ (π β (domaβ(πΌβπ)) = π) | ||
Theorem | idacd 18017 | Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ (π β π β π΅) β β’ (π β (codaβ(πΌβπ)) = π) | ||
Theorem | idaf 18018 | The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ πΌ = (IdaβπΆ) & β’ π΅ = (BaseβπΆ) & β’ (π β πΆ β Cat) & β’ π΄ = (ArrowβπΆ) β β’ (π β πΌ:π΅βΆπ΄) | ||
Theorem | coafval 18019* | The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π΄ = (ArrowβπΆ) & β’ β = (compβπΆ) β β’ Β· = (π β π΄, π β {β β π΄ β£ (codaββ) = (domaβπ)} β¦ β¨(domaβπ), (codaβπ), ((2nd βπ)(β¨(domaβπ), (domaβπ)β© β (codaβπ))(2nd βπ))β©) | ||
Theorem | eldmcoa 18020 | A pair β¨πΊ, πΉβ© is in the domain of the arrow composition, if the domain of πΊ equals the codomain of πΉ. (In this case we say πΊ and πΉ are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π΄ = (ArrowβπΆ) β β’ (πΊdom Β· πΉ β (πΉ β π΄ β§ πΊ β π΄ β§ (codaβπΉ) = (domaβπΊ))) | ||
Theorem | dmcoass 18021 | The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π΄ = (ArrowβπΆ) β β’ dom Β· β (π΄ Γ π΄) | ||
Theorem | homdmcoa 18022 | If πΉ:πβΆπ and πΊ:πβΆπ, then πΊ and πΉ are composable. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π» = (HomaβπΆ) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β πΊdom Β· πΉ) | ||
Theorem | coaval 18023 | Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π» = (HomaβπΆ) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) & β’ β = (compβπΆ) β β’ (π β (πΊ Β· πΉ) = β¨π, π, ((2nd βπΊ)(β¨π, πβ© β π)(2nd βπΉ))β©) | ||
Theorem | coa2 18024 | The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π» = (HomaβπΆ) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) & β’ β = (compβπΆ) β β’ (π β (2nd β(πΊ Β· πΉ)) = ((2nd βπΊ)(β¨π, πβ© β π)(2nd βπΉ))) | ||
Theorem | coahom 18025 | The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π» = (HomaβπΆ) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) β β’ (π β (πΊ Β· πΉ) β (ππ»π)) | ||
Theorem | coapm 18026 | Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ Β· = (compaβπΆ) & β’ π΄ = (ArrowβπΆ) β β’ Β· β (π΄ βpm (π΄ Γ π΄)) | ||
Theorem | arwlid 18027 | Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π» = (HomaβπΆ) & β’ Β· = (compaβπΆ) & β’ 1 = (IdaβπΆ) & β’ (π β πΉ β (ππ»π)) β β’ (π β (( 1 βπ) Β· πΉ) = πΉ) | ||
Theorem | arwrid 18028 | Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π» = (HomaβπΆ) & β’ Β· = (compaβπΆ) & β’ 1 = (IdaβπΆ) & β’ (π β πΉ β (ππ»π)) β β’ (π β (πΉ Β· ( 1 βπ)) = πΉ) | ||
Theorem | arwass 18029 | Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
β’ π» = (HomaβπΆ) & β’ Β· = (compaβπΆ) & β’ 1 = (IdaβπΆ) & β’ (π β πΉ β (ππ»π)) & β’ (π β πΊ β (ππ»π)) & β’ (π β πΎ β (ππ»π)) β β’ (π β ((πΎ Β· πΊ) Β· πΉ) = (πΎ Β· (πΊ Β· πΉ))) | ||
Syntax | csetc 18030 | Extend class notation to include the category Set. |
class SetCat | ||
Definition | df-setc 18031* | Definition of the category Set, relativized to a subset π’. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in π’ and functions between these sets. Generally, we will take π’ to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.) |
β’ SetCat = (π’ β V β¦ {β¨(Baseβndx), π’β©, β¨(Hom βndx), (π₯ β π’, π¦ β π’ β¦ (π¦ βm π₯))β©, β¨(compβndx), (π£ β (π’ Γ π’), π§ β π’ β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))β©}) | ||
Theorem | setcval 18032* | Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π» = (π₯ β π, π¦ β π β¦ (π¦ βm π₯))) & β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))) β β’ (π β πΆ = {β¨(Baseβndx), πβ©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) | ||
Theorem | setcbas 18033 | Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) β β’ (π β π = (BaseβπΆ)) | ||
Theorem | setchomfval 18034* | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) β β’ (π β π» = (π₯ β π, π¦ β π β¦ (π¦ βm π₯))) | ||
Theorem | setchom 18035 | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (ππ»π) = (π βm π)) | ||
Theorem | elsetchom 18036 | A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πΉ β (ππ»π) β πΉ:πβΆπ)) | ||
Theorem | setccofval 18037* | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ Β· = (compβπΆ) β β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β (π§ βm (2nd βπ£)), π β ((2nd βπ£) βm (1st βπ£)) β¦ (π β π)))) | ||
Theorem | setcco 18038 | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ Β· = (compβπΆ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β πΉ:πβΆπ) & β’ (π β πΊ:πβΆπ) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) = (πΊ β πΉ)) | ||
Theorem | setccatid 18039* | Lemma for setccat 18040. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) β β’ (π β π β (πΆ β Cat β§ (IdβπΆ) = (π₯ β π β¦ ( I βΎ π₯)))) | ||
Theorem | setccat 18040 | The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) β β’ (π β π β πΆ β Cat) | ||
Theorem | setcid 18041 | The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ 1 = (IdβπΆ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ( 1 βπ) = ( I βΎ π)) | ||
Theorem | setcmon 18042 | A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ π = (MonoβπΆ) β β’ (π β (πΉ β (πππ) β πΉ:πβ1-1βπ)) | ||
Theorem | setcepi 18043 | An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ πΈ = (EpiβπΆ) & β’ (π β 2o β π) β β’ (π β (πΉ β (ππΈπ) β πΉ:πβontoβπ)) | ||
Theorem | setcsect 18044 | A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ π = (SectβπΆ) β β’ (π β (πΉ(πππ)πΊ β (πΉ:πβΆπ β§ πΊ:πβΆπ β§ (πΊ β πΉ) = ( I βΎ π)))) | ||
Theorem | setcinv 18045 | An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ π = (InvβπΆ) β β’ (π β (πΉ(πππ)πΊ β (πΉ:πβ1-1-ontoβπ β§ πΊ = β‘πΉ))) | ||
Theorem | setciso 18046 | An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ πΌ = (IsoβπΆ) β β’ (π β (πΉ β (ππΌπ) β πΉ:πβ1-1-ontoβπ)) | ||
Theorem | resssetc 18047 | The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCatβπ categories for different π are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ π· = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((Homf β(πΆ βΎs π)) = (Homf βπ·) β§ (compfβ(πΆ βΎs π)) = (compfβπ·))) | ||
Theorem | funcsetcres2 18048 | A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.) |
β’ πΆ = (SetCatβπ) & β’ π· = (SetCatβπ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β (πΈ Func π·) β (πΈ Func πΆ)) | ||
Theorem | setc2obas 18049 | β and 1o are distinct objects in (SetCatβ2o). This combined with setc2ohom 18050 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17617 and cat1 18052. (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ πΆ = (SetCatβ2o) & β’ π΅ = (BaseβπΆ) β β’ (β β π΅ β§ 1o β π΅ β§ 1o β β ) | ||
Theorem | setc2ohom 18050 | (SetCatβ2o) is a category (provable from setccat 18040 and 2oex 8480) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18049. Notably, the empty set β is simultaneously an object (setc2obas 18049), an identity morphism from β to β (setcid 18041 or thincid 47742), and a non-identity morphism from β to 1o. See cat1lem 18051 and cat1 18052 for a more general statement. This category is also thin (setc2othin 47765), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 47763 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.) |
β’ πΆ = (SetCatβ2o) & β’ π» = (Hom βπΆ) β β’ β β ((β π»β ) β© (β π»1o)) | ||
Theorem | cat1lem 18051* | The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 18052. (Contributed by Zhi Wang, 15-Sep-2024.) |
β’ πΆ = (SetCatβπ) & β’ (π β π β π) & β’ π΅ = (BaseβπΆ) & β’ π» = (Hom βπΆ) & β’ (π β β β π) & β’ (π β π β π) & β’ (π β β β π) β β’ (π β βπ₯ β π΅ βπ¦ β π΅ βπ§ β π΅ βπ€ β π΅ (((π₯π»π¦) β© (π§π»π€)) β β β§ Β¬ (π₯ = π§ β§ π¦ = π€))) | ||
Theorem | cat1 18052* | The definition of category df-cat 17617 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18049 and setc2ohom 18050 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17981 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.) |
β’ βπ β Cat [(Baseβπ) / π][(Hom βπ) / β] Β¬ βπ₯ β π βπ¦ β π βπ§ β π βπ€ β π (((π₯βπ¦) β© (π§βπ€)) β β β (π₯ = π§ β§ π¦ = π€)) | ||
Syntax | ccatc 18053 | Extend class notation to include the category Cat. |
class CatCat | ||
Definition | df-catc 18054* | Definition of the category Cat, which consists of all categories in the universe π’ (i.e., "π’-small categories", see Definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "π’ -large categories", which can be expressed as (Cat β π’). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ CatCat = (π’ β V β¦ β¦(π’ β© Cat) / πβ¦{β¨(Baseβndx), πβ©, β¨(Hom βndx), (π₯ β π, π¦ β π β¦ (π₯ Func π¦))β©, β¨(compβndx), (π£ β (π Γ π), π§ β π β¦ (π β ((2nd βπ£) Func π§), π β ( Func βπ£) β¦ (π βfunc π)))β©}) | ||
Theorem | catcval 18055* | Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ (π β π β π) & β’ (π β π΅ = (π β© Cat)) & β’ (π β π» = (π₯ β π΅, π¦ β π΅ β¦ (π₯ Func π¦))) & β’ (π β Β· = (π£ β (π΅ Γ π΅), π§ β π΅ β¦ (π β ((2nd βπ£) Func π§), π β ( Func βπ£) β¦ (π βfunc π)))) β β’ (π β πΆ = {β¨(Baseβndx), π΅β©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) | ||
Theorem | catcbas 18056 | Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π) β β’ (π β π΅ = (π β© Cat)) | ||
Theorem | catchomfval 18057* | Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) β β’ (π β π» = (π₯ β π΅, π¦ β π΅ β¦ (π₯ Func π¦))) | ||
Theorem | catchom 18058 | Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β (ππ»π) = (π Func π)) | ||
Theorem | catccofval 18059* | Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π) & β’ Β· = (compβπΆ) β β’ (π β Β· = (π£ β (π΅ Γ π΅), π§ β π΅ β¦ (π β ((2nd βπ£) Func π§), π β ( Func βπ£) β¦ (π βfunc π)))) | ||
Theorem | catcco 18060 | Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β π) & β’ Β· = (compβπΆ) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β πΉ β (π Func π)) & β’ (π β πΊ β (π Func π)) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) = (πΊ βfunc πΉ)) | ||
Theorem | catccatid 18061* | Lemma for catccat 18063. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) β β’ (π β π β (πΆ β Cat β§ (IdβπΆ) = (π₯ β π΅ β¦ (idfuncβπ₯)))) | ||
Theorem | catcid 18062 | The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ 1 = (IdβπΆ) & β’ πΌ = (idfuncβπ) & β’ (π β π β π) & β’ (π β π β π΅) β β’ (π β ( 1 βπ) = πΌ) | ||
Theorem | catccat 18063 | The category of categories is a category, see remark 3.48 in [Adamek] p. 40. (Clearly it cannot be an element of itself, hence it is "π -large".) (Contributed by Mario Carneiro, 3-Jan-2017.) |
β’ πΆ = (CatCatβπ) β β’ (π β π β πΆ β Cat) | ||
Theorem | resscatc 18064 | The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCatβπ categories for different π are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π· = (CatCatβπ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ((Homf β(πΆ βΎs π)) = (Homf βπ·) β§ (compfβ(πΆ βΎs π)) = (compfβπ·))) | ||
Theorem | catcisolem 18065* | Lemma for catciso 18066. (Contributed by Mario Carneiro, 29-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (Baseβπ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ πΌ = (InvβπΆ) & β’ π» = (π₯ β π, π¦ β π β¦ β‘((β‘πΉβπ₯)πΊ(β‘πΉβπ¦))) & β’ (π β πΉ((π Full π) β© (π Faith π))πΊ) & β’ (π β πΉ:π β1-1-ontoβπ) β β’ (π β β¨πΉ, πΊβ©(ππΌπ)β¨β‘πΉ, π»β©) | ||
Theorem | catciso 18066 | A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (Baseβπ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ πΌ = (IsoβπΆ) β β’ (π β (πΉ β (ππΌπ) β (πΉ β ((π Full π) β© (π Faith π)) β§ (1st βπΉ):π β1-1-ontoβπ))) | ||
Theorem | catcbascl 18067 | An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18072. (Contributed by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) & β’ (π β π β π΅) β β’ (π β π β π) | ||
Theorem | catcslotelcl 18068 | A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18072. (Contributed by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) & β’ (π β π β π΅) & β’ πΈ = Slot (πΈβndx) β β’ (π β (πΈβπ) β π) | ||
Theorem | catcbaselcl 18069 | The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18072. (Contributed by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) & β’ (π β π β π΅) β β’ (π β (Baseβπ) β π) | ||
Theorem | catchomcl 18070 | The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18072. (Contributed by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) & β’ (π β π β π΅) β β’ (π β (Hom βπ) β π) | ||
Theorem | catcccocl 18071 | The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18072. (Contributed by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) & β’ (π β π β π΅) β β’ (π β (compβπ) β π) | ||
Theorem | catcoppccl 18072 | The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (oppCatβπ) & β’ (π β π β WUni) & β’ (π β Ο β π) & β’ (π β π β π΅) β β’ (π β π β π΅) | ||
Theorem | catcoppcclOLD 18073 | Obsolete proof of catcoppccl 18072 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (oppCatβπ) & β’ (π β π β WUni) & β’ (π β Ο β π) & β’ (π β π β π΅) β β’ (π β π β π΅) | ||
Theorem | catcfuccl 18074 | The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (π FuncCat π) & β’ (π β π β WUni) & β’ (π β Ο β π) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β π β π΅) | ||
Theorem | catcfucclOLD 18075 | Obsolete proof of catcfuccl 18074 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΆ = (CatCatβπ) & β’ π΅ = (BaseβπΆ) & β’ π = (π FuncCat π) & β’ (π β π β WUni) & β’ (π β Ο β π) & β’ (π β π β π΅) & β’ (π β π β π΅) β β’ (π β π β π΅) | ||
The "category of extensible structures" ExtStrCat is the category of all sets in a universe regarded as extensible structures and the functions between their base sets, see df-estrc 18079. Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are all sets in a universe π’, which can be an arbitrary set, see estrcbas 18081. Generally, we will take π’ to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 18077 we do not need to restrict the universe to sets which "have a base". The morphisms (or arrows) between two objects, i.e. sets from the universe, are the mappings between their base sets, see estrchomfval 18082, whereas the composition is the ordinary composition of functions, see estrccofval 18085 and estrcco 18086. It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 18089 with the identity function as identity arrow, see estrcid 18090. In the following, some background information about the category of extensible structures is given, taken from the discussion in Github issue #1507 (see https://github.com/metamath/set.mm/issues/1507 18090): At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordereds pairs, see dfrngc2 46960 and dfringc2 47006, but with special compositions). With this definitions, however, Theorem rngcresringcat 47018 could not be proven, because the compositions were not compatible. Unfortunately, no precise definition of the composition within the category of rings could be found in the literature. In section 3.3 EXAMPLES, paragraph (2) of [Adamek] p. 22, however, a definition is given for "Grp", the category of groups: "The following constructs; i.e., categories of structured sets and structure-preserving functions between them (o will always be the composition of functions and idA will always be the identity function on A): ... (b) Grp with objects all groups and morphisms all homomorphisms between them." Therefore, the compositions should have been harmonized by using the composition of the category of sets SetCat, see df-setc 18031, which is the ordinary composition of functions. Analogously, categories of Rngs (and Rings) could have been shown to be restrictions resp. subcategories of the category of sets. BJ and MC observed, however, that "... βΎcat [cannot be used] to restrict the category Set to Ring, because the homs are different. Although Ring is a concrete category, a hom between rings R and S is a function (Base`R) --> (Base`S) with certain properties, unlike in Set where it is a function R --> S.". Therefore, MC suggested that "we could have an alternative version of the Set category consisting of extensible structures (in U) together with (A Hom B) := (Base`A) --> (Base`B). This category is not isomorphic to Set because different extensible structures can have the same base set, but it is equivalent to Set; the relevant functors are (U`A) = (Base`A), the forgetful functor, and (F`A) = { <. (Base`ndx), A >. }". This led to the current definition of ExtStrCat, see df-estrc 18079. The claimed equivalence is proven by equivestrcsetc 18109. Having a definition of a category of extensible structures, the categories of non-unital and unital rings can be defined as appropriate restrictions of the category of extensible structures, see df-rngc 46947 and df-ringc 46993. In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 47018, although the morphisms of the shown categories are different ( "->" means "is subcategory of"): RingCat-> RngCat-> GrpCat -> MndCat -> MgmCat -> ExtStrCat According to MC, "If we generalize from subcategories to embeddings, then we can even fit SetCat into the chain, equivalent to ExtStrCat at the end." As mentioned before, the equivalence of SetCat and ExtStrCat is proven by equivestrcsetc 18109. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 18124. Remark: equivestrcsetc 18109 as well as embedsetcestrc 18124 require that the index of the base set extractor is contained within the considered universe. This is ensured by assuming that the natural numbers are contained within the considered universe: Ο β π (see wunndx 17133), but it would be currently sufficient to assume that 1 β π, because the index value of the base set extractor is hard-coded as 1, see basendx 17158. Some people, however, feel uncomfortable to say that a ring "is a" group (without mentioning the restriction to the addition, which is usually found in the literature, e.g., the definition of a ring in [Herstein] p. 126: "... Note that so far all we have said is that R is an abelian group under +.". The main argument against a ring being a group is the number of components/slots: usually, a group consists of (exactly!) two components (a base set and an operation), whereas a ring consists of (exactly!) three components (a base set and two operations). According to this "definition", a ring cannot be a group. This is also an (unfortunately informal) argument for the category of rings not being a subcategory of the category of abelian groups in "Categories and Functors", Bodo Pareigis, Academic Press, New York, London, 1970: "A category A is called a subcategory of a category B if Ob(A) β Ob(B) and MorA(X,Y) β MorB(X,Y) for all X,Y e. Ob(A), if the composition of morphisms in A coincides with the composition of the same morphisms in B and if the identity of an object in A is also the identity of the same object viewed as an object in B. Then there is a forgetful functor from A to B. We note that Ri [the category of rings] is not a subcategory of Ab [the category of abelian groups]. In fact, Ob(Ri) β Ob(Ab) is not true, although every ring can also be regarded as an abelian group. The corresponding abelian groups of two rings may coincide even if the rings do not coincide. The multiplication may be defined differently.". As long as we define Rings, Groups, etc. in a way that π΄ β Ring β π΄ β Grp is valid (see ringgrp 20133) the corresponding categories are in a subcategory relation. If we do not want Rings to be Groups (then the category of rings would not be a subcategory of the category of groups, as observed by Pareigis), we would have to change the definitions of Magmas, Monoids, Groups, Rings etc. to restrict them to have exactly the required number of slots, so that the following holds π β Grp β π Struct β¨(Baseβndx), (+gβndx)β© π β Ring β π Struct β¨(Baseβndx), (+gβndx), (.rβndx)β© | ||
Theorem | fncnvimaeqv 18076 | The inverse images of the universal class V under functions on the universal class V are the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
β’ (πΉ Fn V β (β‘πΉ β V) = V) | ||
Theorem | bascnvimaeqv 18077 | The inverse image of the universal class V under the base function is the universal class V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
β’ (β‘Base β V) = V | ||
Syntax | cestrc 18078 | Extend class notation to include the category ExtStr. |
class ExtStrCat | ||
Definition | df-estrc 18079* | Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe π’ regarded as extensible structures and whose morphisms are the functions between their base sets. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 18077 we do not need to restrict the universe to sets which "have a base". Generally, we will take π’ to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Proposed by Mario Carneiro, 5-Mar-2020.) (Contributed by AV, 7-Mar-2020.) |
β’ ExtStrCat = (π’ β V β¦ {β¨(Baseβndx), π’β©, β¨(Hom βndx), (π₯ β π’, π¦ β π’ β¦ ((Baseβπ¦) βm (Baseβπ₯)))β©, β¨(compβndx), (π£ β (π’ Γ π’), π§ β π’ β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))β©}) | ||
Theorem | estrcval 18080* | Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ (π β π» = (π₯ β π, π¦ β π β¦ ((Baseβπ¦) βm (Baseβπ₯)))) & β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))) β β’ (π β πΆ = {β¨(Baseβndx), πβ©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) | ||
Theorem | estrcbas 18081 | Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) β β’ (π β π = (BaseβπΆ)) | ||
Theorem | estrchomfval 18082* | Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) β β’ (π β π» = (π₯ β π, π¦ β π β¦ ((Baseβπ¦) βm (Baseβπ₯)))) | ||
Theorem | estrchom 18083 | The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) & β’ (π β π β π) & β’ (π β π β π) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) β β’ (π β (ππ»π) = (π΅ βm π΄)) | ||
Theorem | elestrchom 18084 | A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) & β’ (π β π β π) & β’ (π β π β π) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) β β’ (π β (πΉ β (ππ»π) β πΉ:π΄βΆπ΅)) | ||
Theorem | estrccofval 18085* | Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ Β· = (compβπΆ) β β’ (π β Β· = (π£ β (π Γ π), π§ β π β¦ (π β ((Baseβπ§) βm (Baseβ(2nd βπ£))), π β ((Baseβ(2nd βπ£)) βm (Baseβ(1st βπ£))) β¦ (π β π)))) | ||
Theorem | estrcco 18086 | Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ Β· = (compβπΆ) & β’ (π β π β π) & β’ (π β π β π) & β’ (π β π β π) & β’ π΄ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ π· = (Baseβπ) & β’ (π β πΉ:π΄βΆπ΅) & β’ (π β πΊ:π΅βΆπ·) β β’ (π β (πΊ(β¨π, πβ© Β· π)πΉ) = (πΊ β πΉ)) | ||
Theorem | estrcbasbas 18087 | An element of the base set of the base set of the category of extensible structures (i.e. the base set of an extensible structure) belongs to the considered weak universe. (Contributed by AV, 22-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ π΅ = (BaseβπΆ) & β’ (π β π β WUni) β β’ ((π β§ πΈ β π΅) β (BaseβπΈ) β π) | ||
Theorem | estrccatid 18088* | Lemma for estrccat 18089. (Contributed by AV, 8-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) β β’ (π β π β (πΆ β Cat β§ (IdβπΆ) = (π₯ β π β¦ ( I βΎ (Baseβπ₯))))) | ||
Theorem | estrccat 18089 | The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) β β’ (π β π β πΆ β Cat) | ||
Theorem | estrcid 18090 | The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ 1 = (IdβπΆ) & β’ (π β π β π) & β’ (π β π β π) β β’ (π β ( 1 βπ) = ( I βΎ (Baseβπ))) | ||
Theorem | estrchomfn 18091 | The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) β β’ (π β π» Fn (π Γ π)) | ||
Theorem | estrchomfeqhom 18092 | The functionalized Hom-set operation equals the Hom-set operation in the category of extensible structures (in a universe). (Contributed by AV, 8-Mar-2020.) |
β’ πΆ = (ExtStrCatβπ) & β’ (π β π β π) & β’ π» = (Hom βπΆ) β β’ (π β (Homf βπΆ) = π») | ||
Theorem | estrreslem1 18093 | Lemma 1 for estrres 18096. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.) |
β’ (π β πΆ = {β¨(Baseβndx), π΅β©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) & β’ (π β π΅ β π) β β’ (π β π΅ = (BaseβπΆ)) | ||
Theorem | estrreslem1OLD 18094 | Obsolete version of estrreslem1 18093 as of 28-Oct-2024. Lemma 1 for estrres 18096. (Contributed by AV, 14-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β πΆ = {β¨(Baseβndx), π΅β©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) & β’ (π β π΅ β π) β β’ (π β π΅ = (BaseβπΆ)) | ||
Theorem | estrreslem2 18095 | Lemma 2 for estrres 18096. (Contributed by AV, 14-Mar-2020.) |
β’ (π β πΆ = {β¨(Baseβndx), π΅β©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) & β’ (π β π΅ β π) & β’ (π β π» β π) & β’ (π β Β· β π) β β’ (π β (Baseβndx) β dom πΆ) | ||
Theorem | estrres 18096 | Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.) (Revised by AV, 3-Jul-2022.) |
β’ (π β πΆ = {β¨(Baseβndx), π΅β©, β¨(Hom βndx), π»β©, β¨(compβndx), Β· β©}) & β’ (π β π΅ β π) & β’ (π β π» β π) & β’ (π β Β· β π) & β’ (π β πΊ β π) & β’ (π β π΄ β π΅) β β’ (π β ((πΆ βΎs π΄) sSet β¨(Hom βndx), πΊβ©) = {β¨(Baseβndx), π΄β©, β¨(Hom βndx), πΊβ©, β¨(compβndx), Β· β©}) | ||
Theorem | funcestrcsetclem1 18097* | Lemma 1 for funcestrcsetc 18106. (Contributed by AV, 22-Mar-2020.) |
β’ πΈ = (ExtStrCatβπ) & β’ π = (SetCatβπ) & β’ π΅ = (BaseβπΈ) & β’ πΆ = (Baseβπ) & β’ (π β π β WUni) & β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) β β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) | ||
Theorem | funcestrcsetclem2 18098* | Lemma 2 for funcestrcsetc 18106. (Contributed by AV, 22-Mar-2020.) |
β’ πΈ = (ExtStrCatβπ) & β’ π = (SetCatβπ) & β’ π΅ = (BaseβπΈ) & β’ πΆ = (Baseβπ) & β’ (π β π β WUni) & β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) β β’ ((π β§ π β π΅) β (πΉβπ) β π) | ||
Theorem | funcestrcsetclem3 18099* | Lemma 3 for funcestrcsetc 18106. (Contributed by AV, 22-Mar-2020.) |
β’ πΈ = (ExtStrCatβπ) & β’ π = (SetCatβπ) & β’ π΅ = (BaseβπΈ) & β’ πΆ = (Baseβπ) & β’ (π β π β WUni) & β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) β β’ (π β πΉ:π΅βΆπΆ) | ||
Theorem | funcestrcsetclem4 18100* | Lemma 4 for funcestrcsetc 18106. (Contributed by AV, 22-Mar-2020.) |
β’ πΈ = (ExtStrCatβπ) & β’ π = (SetCatβπ) & β’ π΅ = (BaseβπΈ) & β’ πΆ = (Baseβπ) & β’ (π β π β WUni) & β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) & β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) β β’ (π β πΊ Fn (π΅ Γ π΅)) |
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