P. 168 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	arwhom 16701	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
|- A = (Nat ` C )   &    |- J = ( Hom ` C )   =>    |- ( F e. A -> ( 2nd ` F ) e. ( (domA ` F ) J (coda ` F ) ) )
Theorem	arwdmcd 16702	Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- A = (Nat ` C )   =>    |- ( F e. A -> F = <. (domA ` F ) , (coda ` F ) , ( 2nd ` F ) >. )
8.2.1  Identity and composition for arrows
Syntax	cida 16703	Extend class notation to include identity for arrows.
class Ida
Syntax	ccoa 16704	Extend class notation to include composition for arrows.
class compa
Definition	df-ida 16705*	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.)
|- Ida = ( c e. Cat |-> ( x e. ( Base ` c ) |-> <. x , x , ( ( Id ` c ) ` x ) >. ) )
Definition	df-coa 16706*	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.)
|- compa = ( c e. Cat |-> ( g e. (Nat ` c ) , f e. { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) >. ) )
Theorem	idafval 16707*	Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   =>    |- ( ph -> I = ( x e. B |-> <. x , x , ( .1. ` x ) >. ) )
Theorem	idaval 16708	Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( I ` X ) = <. X , X , ( .1. ` X ) >. )
Theorem	ida2 16709	Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( 2nd ` ( I ` X ) ) = ( .1. ` X ) )
Theorem	idahom 16710	Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- H = (Homa ` C )   =>    |- ( ph -> ( I ` X ) e. ( X H X ) )
Theorem	idadm 16711	Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> (domA ` ( I ` X ) ) = X )
Theorem	idacd 16712	Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> (coda ` ( I ` X ) ) = X )
Theorem	idaf 16713	The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- I = (Ida ` C )   &    |- B = ( Base ` C )   &    |- ( ph -> C e. Cat )   &    |- A = (Nat ` C )   =>    |- ( ph -> I : B --> A )
Theorem	coafval 16714*	The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- A = (Nat ` C )   &    |- .xb = (comp ` C )   =>    |- .x. = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )
Theorem	eldmcoa 16715	A pair <. G , F >. is in the domain of the arrow composition, if the domain of G equals the codomain of F. (In this case we say G and F are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- A = (Nat ` C )   =>    |- ( G dom .x. F <-> ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) )
Theorem	dmcoass 16716	The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- A = (Nat ` C )   =>    |- dom .x. C_ ( A X. A )
Theorem	homdmcoa 16717	If F : X --> Y and G : Y --> Z, then G and F are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- H = (Homa ` C )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> G dom .x. F )
Theorem	coaval 16718	Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- H = (Homa ` C )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   &    |- .xb = (comp ` C )   =>    |- ( ph -> ( G .x. F ) = <. X , Z , ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) >. )
Theorem	coa2 16719	The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- H = (Homa ` C )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   &    |- .xb = (comp ` C )   =>    |- ( ph -> ( 2nd ` ( G .x. F ) ) = ( ( 2nd ` G ) ( <. X , Y >. .xb Z ) ( 2nd ` F ) ) )
Theorem	coahom 16720	The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- H = (Homa ` C )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G .x. F ) e. ( X H Z ) )
Theorem	coapm 16721	Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- .x. = (compa ` C )   &    |- A = (Nat ` C )   =>    |- .x. e. ( A ^pm ( A X. A ) )
Theorem	arwlid 16722	Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- H = (Homa ` C )   &    |- .x. = (compa ` C )   &    |- .1. = (Ida ` C )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( ( .1. ` Y ) .x. F ) = F )
Theorem	arwrid 16723	Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- H = (Homa ` C )   &    |- .x. = (compa ` C )   &    |- .1. = (Ida ` C )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( F .x. ( .1. ` X ) ) = F )
Theorem	arwass 16724	Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- H = (Homa ` C )   &    |- .x. = (compa ` C )   &    |- .1. = (Ida ` C )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   &    |- ( ph -> K e. ( Z H W ) )   =>    |- ( ph -> ( ( K .x. G ) .x. F ) = ( K .x. ( G .x. F ) ) )
8.3  Examples of categories
8.3.1  The category of sets
Syntax	csetc 16725	Extend class notation to include the category Set.
class SetCat
Definition	df-setc 16726*	Definition of the category Set, relativized to a subset u. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in u and functions between these sets. Generally, we will take u 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.)
|- SetCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. } )
Theorem	setcval 16727*	Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> H = ( x e. U , y e. U |-> ( y ^m x ) ) )   &    |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	setcbas 16728	Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   =>    |- ( ph -> U = ( Base ` C ) )
Theorem	setchomfval 16729*	Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( x e. U , y e. U |-> ( y ^m x ) ) )
Theorem	setchom 16730	Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   =>    |- ( ph -> ( X H Y ) = ( Y ^m X ) )
Theorem	elsetchom 16731	A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   =>    |- ( ph -> ( F e. ( X H Y ) <-> F : X --> Y ) )
Theorem	setccofval 16732*	Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) )
Theorem	setcco 16733	Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- ( ph -> Z e. U )   &    |- ( ph -> F : X --> Y )   &    |- ( ph -> G : Y --> Z )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )
Theorem	setccatid 16734*	Lemma for setccat 16735. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   =>    |- ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` x ) ) ) )
Theorem	setccat 16735	The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	setcid 16736	The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- .1. = ( Id ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   =>    |- ( ph -> ( .1. ` X ) = ( _I |` X ) )
Theorem	setcmon 16737	A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- M = (Mono ` C )   =>    |- ( ph -> ( F e. ( X M Y ) <-> F : X -1-1-> Y ) )
Theorem	setcepi 16738	An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- E = (Epi ` C )   &    |- ( ph -> 2o e. U )   =>    |- ( ph -> ( F e. ( X E Y ) <-> F : X -onto-> Y ) )
Theorem	setcsect 16739	A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- S = (Sect ` C )   =>    |- ( ph -> ( F ( X S Y ) G <-> ( F : X --> Y /\ G : Y --> X /\ ( G o. F ) = ( _I |` X ) ) ) )
Theorem	setcinv 16740	An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- N = (Inv ` C )   =>    |- ( ph -> ( F ( X N Y ) G <-> ( F : X -1-1-onto-> Y /\ G = `' F ) ) )
Theorem	setciso 16741	An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = ( SetCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- I = ( Iso ` C )   =>    |- ( ph -> ( F e. ( X I Y ) <-> F : X -1-1-onto-> Y ) )
Theorem	resssetc 16742	The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat ` U categories for different U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- C = ( SetCat ` U )   &    |- D = ( SetCat ` V )   &    |- ( ph -> U e. W )   &    |- ( ph -> V C_ U )   =>    |- ( ph -> ( ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) /\ (compf ` ( C ↾s V ) ) = (compf ` D ) ) )
Theorem	funcsetcres2 16743	A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
|- C = ( SetCat ` U )   &    |- D = ( SetCat ` V )   &    |- ( ph -> U e. W )   &    |- ( ph -> V C_ U )   =>    |- ( ph -> ( E Func D ) C_ ( E Func C ) )
8.3.2  The category of categories
Syntax	ccatc 16744	Extend class notation to include the category Cat.
class CatCat
Definition	df-catc 16745*	Definition of the category Cat, which consists of all categories in the universe u (i.e. "small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- CatCat = ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. (comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } )
Theorem	catcval 16746*	Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> B = ( U i^i Cat ) )   &    |- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )   &    |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	catcbas 16747	Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   =>    |- ( ph -> B = ( U i^i Cat ) )
Theorem	catchomfval 16748*	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
Theorem	catchom 16749	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X H Y ) = ( X Func Y ) )
Theorem	catccofval 16750*	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = ( v e. ( B X. B ) , z e. B |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) )
Theorem	catcco 16751	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X Func Y ) )   &    |- ( ph -> G e. ( Y Func Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o.func F ) )
Theorem	catccatid 16752*	Lemma for catccat 16754. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   =>    |- ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. B |-> (idfunc ` x ) ) ) )
Theorem	catcid 16753	The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- .1. = ( Id ` C )   &    |- I = (idfunc ` X )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .1. ` X ) = I )
Theorem	catccat 16754	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 "large" with respect to U.) (Contributed by Mario Carneiro, 3-Jan-2017.)
|- C = (CatCat ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	resscatc 16755	The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat ` U categories for different U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- C = (CatCat ` U )   &    |- D = (CatCat ` V )   &    |- ( ph -> U e. W )   &    |- ( ph -> V C_ U )   =>    |- ( ph -> ( ( Hom f ` ( C ↾s V ) ) = ( Hom f ` D ) /\ (compf ` ( C ↾s V ) ) = (compf ` D ) ) )
Theorem	catcisolem 16756*	Lemma for catciso 16757. (Contributed by Mario Carneiro, 29-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- R = ( Base ` X )   &    |- S = ( Base ` Y )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- I = (Inv ` C )   &    |- H = ( x e. S , y e. S |-> `' ( ( `' F ` x ) G ( `' F ` y ) ) )   &    |- ( ph -> F ( ( X Full Y ) i^i ( X Faith Y ) ) G )   &    |- ( ph -> F : R -1-1-onto-> S )   =>    |- ( ph -> <. F , G >. ( X I Y ) <. `' F , H >. )
Theorem	catciso 16757	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.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- R = ( Base ` X )   &    |- S = ( Base ` Y )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- I = ( Iso ` C )   =>    |- ( ph -> ( F e. ( X I Y ) <-> ( F e. ( ( X Full Y ) i^i ( X Faith Y ) ) /\ ( 1st ` F ) : R -1-1-onto-> S ) ) )
Theorem	catcoppccl 16758	The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- O = (oppCat ` X )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- ( ph -> X e. B )   =>    |- ( ph -> O e. B )
Theorem	catcfuccl 16759	The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- C = (CatCat ` U )   &    |- B = ( Base ` C )   &    |- Q = ( X FuncCat Y )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> Q e. B )
8.3.3  The category of extensible structures 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 16763. 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 u, which can be an arbitrary set, see estrcbas 16765. Generally, we will take u 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 16761 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 16766, whereas the composition is the ordinary composition of functions, see estrccofval 16769 and estrcco 16770. It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 16773 with the identity function as identity arrow, see estrcid 16774. 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): 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 41972 and dfringc2 42018, but with special compositions). With this definitions, however, theorem rngcresringcat 42030 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 16726, 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 "... |`cat [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 16763. The claimed equivalence is proven by equivestrcsetc 16792. 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 41959 and df-ringc 42005. In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 42030, 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 16792. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 16807. Remark: equivestrcsetc 16792 as well as embedsetcestrc 16807 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: om e. U (see wunndx 15878), but it would be currently sufficient to assume that 1 e. U, because the index value of the base set extractor is hard-coded as 1, see basendx 15923. 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) C_ Ob(B) and MorA(X,Y) C_ 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) C_ 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 A e. Ring -> A e. Grp is valid (see ringgrp 18552) 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 g e. Grp -> g Struct <. ( Base ` ndx ) , ( +g ` ndx ) >. r e. Ring -> r Struct <. ( Base ` ndx ) , ( +g ` ndx ) , ( .r ` ndx ) >.
Theorem	fncnvimaeqv 16760	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.)
|- ( F Fn _V -> ( `' F " _V ) = _V )
Theorem	bascnvimaeqv 16761	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.)
|- ( `' Base " _V ) = _V
Syntax	cestrc 16762	Extend class notation to include the category ExtStr.
class ExtStrCat
Definition	df-estrc 16763*	Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe u 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 16761 we do not need to restrict the universe to sets which "have a base". Generally, we will take u 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.)
|- ExtStrCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) >. , <. (comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) >. } )
Theorem	estrcval 16764*	Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- ( ph -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )   &    |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )   =>    |- ( ph -> C = { <. ( Base ` ndx ) , U >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
Theorem	estrcbas 16765	Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   =>    |- ( ph -> U = ( Base ` C ) )
Theorem	estrchomfval 16766*	Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) )
Theorem	estrchom 16767	The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- A = ( Base ` X )   &    |- B = ( Base ` Y )   =>    |- ( ph -> ( X H Y ) = ( B ^m A ) )
Theorem	elestrchom 16768	A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- A = ( Base ` X )   &    |- B = ( Base ` Y )   =>    |- ( ph -> ( F e. ( X H Y ) <-> F : A --> B ) )
Theorem	estrccofval 16769*	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   =>    |- ( ph -> .x. = ( v e. ( U X. U ) , z e. U |-> ( g e. ( ( Base ` z ) ^m ( Base ` ( 2nd ` v ) ) ) , f e. ( ( Base ` ( 2nd ` v ) ) ^m ( Base ` ( 1st ` v ) ) ) |-> ( g o. f ) ) ) )
Theorem	estrcco 16770	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- ( ph -> Z e. U )   &    |- A = ( Base ` X )   &    |- B = ( Base ` Y )   &    |- D = ( Base ` Z )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : B --> D )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )
Theorem	estrcbasbas 16771	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.)
|- C = (ExtStrCat ` U )   &    |- B = ( Base ` C )   &    |- ( ph -> U e. WUni )   =>    |- ( ( ph /\ E e. B ) -> ( Base ` E ) e. U )
Theorem	estrccatid 16772*	Lemma for estrccat 16773. (Contributed by AV, 8-Mar-2020.)
|- C = (ExtStrCat ` U )   =>    |- ( U e. V -> ( C e. Cat /\ ( Id ` C ) = ( x e. U |-> ( _I |` ( Base ` x ) ) ) ) )
Theorem	estrccat 16773	The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
|- C = (ExtStrCat ` U )   =>    |- ( U e. V -> C e. Cat )
Theorem	estrcid 16774	The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- .1. = ( Id ` C )   &    |- ( ph -> U e. V )   &    |- ( ph -> X e. U )   =>    |- ( ph -> ( .1. ` X ) = ( _I |` ( Base ` X ) ) )
Theorem	estrchomfn 16775	The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> H Fn ( U X. U ) )
Theorem	estrchomfeqhom 16776	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.)
|- C = (ExtStrCat ` U )   &    |- ( ph -> U e. V )   &    |- H = ( Hom ` C )   =>    |- ( ph -> ( Hom f ` C ) = H )
Theorem	estrreslem1 16777	Lemma 1 for estrres 16779. (Contributed by AV, 14-Mar-2020.)
|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   =>    |- ( ph -> B = ( Base ` C ) )
Theorem	estrreslem2 16778	Lemma 2 for estrres 16779. (Contributed by AV, 14-Mar-2020.)
|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> H e. X )   &    |- ( ph -> .x. e. Y )   =>    |- ( ph -> ( Base ` ndx ) e. dom C )
Theorem	estrres 16779	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.)
|- ( ph -> C = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> H e. X )   &    |- ( ph -> .x. e. Y )   &    |- ( ph -> A e. U )   &    |- ( ph -> G e. W )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( ( C ↾s A ) sSet <. ( Hom ` ndx ) , G >. ) = { <. ( Base ` ndx ) , A >. , <. ( Hom ` ndx ) , G >. , <. (comp ` ndx ) , .x. >. } )
Theorem	funcestrcsetclem1 16780*	Lemma 1 for funcestrcsetc 16789. (Contributed by AV, 22-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   =>    |- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )
Theorem	funcestrcsetclem2 16781*	Lemma 2 for funcestrcsetc 16789. (Contributed by AV, 22-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   =>    |- ( ( ph /\ X e. B ) -> ( F ` X ) e. U )
Theorem	funcestrcsetclem3 16782*	Lemma 3 for funcestrcsetc 16789. (Contributed by AV, 22-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   =>    |- ( ph -> F : B --> C )
Theorem	funcestrcsetclem4 16783*	Lemma 4 for funcestrcsetc 16789. (Contributed by AV, 22-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ph -> G Fn ( B X. B ) )
Theorem	funcestrcsetclem5 16784*	Lemma 5 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   &    |- M = ( Base ` X )   &    |- N = ( Base ` Y )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( N ^m M ) ) )
Theorem	funcestrcsetclem6 16785*	Lemma 6 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   &    |- M = ( Base ` X )   &    |- N = ( Base ` Y )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = H )
Theorem	funcestrcsetclem7 16786*	Lemma 7 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ( ph /\ X e. B ) -> ( ( X G X ) ` ( ( Id ` E ) ` X ) ) = ( ( Id ` S ) ` ( F ` X ) ) )
Theorem	funcestrcsetclem8 16787*	Lemma 8 for funcestrcsetc 16789. (Contributed by AV, 15-Feb-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) : ( X ( Hom ` E ) Y ) --> ( ( F ` X ) ( Hom ` S ) ( F ` Y ) ) )
Theorem	funcestrcsetclem9 16788*	Lemma 9 for funcestrcsetc 16789. (Contributed by AV, 23-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ( ph /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( H e. ( X ( Hom ` E ) Y ) /\ K e. ( Y ( Hom ` E ) Z ) ) ) -> ( ( X G Z ) ` ( K ( <. X , Y >. (comp ` E ) Z ) H ) ) = ( ( ( Y G Z ) ` K ) ( <. ( F ` X ) , ( F ` Y ) >. (comp ` S ) ( F ` Z ) ) ( ( X G Y ) ` H ) ) )
Theorem	funcestrcsetc 16789*	The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ph -> F ( E Func S ) G )
Theorem	fthestrcsetc 16790*	The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is faithful. (Contributed by AV, 2-Apr-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ph -> F ( E Faith S ) G )
Theorem	fullestrcsetc 16791*	The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   =>    |- ( ph -> F ( E Full S ) G )
Theorem	equivestrcsetc 16792*	The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (Contributed by AV, 2-Apr-2020.)
|- E = (ExtStrCat ` U )   &    |- S = ( SetCat ` U )   &    |- B = ( Base ` E )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )   &    |- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )   &    |- ( ph -> ( Base ` ndx ) e. U )   =>    |- ( ph -> ( F ( E Faith S ) G /\ F ( E Full S ) G /\ A. b e. C E. a e. B E. i i : b -1-1-onto-> ( F ` a ) ) )
Theorem	setc1strwun 16793	A constructed one-slot structure with the objects of the category of sets as base set in a weak universe. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   =>    |- ( ( ph /\ X e. C ) -> { <. ( Base ` ndx ) , X >. } e. U )
Theorem	funcsetcestrclem1 16794*	Lemma 1 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   =>    |- ( ( ph /\ X e. C ) -> ( F ` X ) = { <. ( Base ` ndx ) , X >. } )
Theorem	funcsetcestrclem2 16795*	Lemma 2 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   =>    |- ( ( ph /\ X e. C ) -> ( F ` X ) e. U )
Theorem	funcsetcestrclem3 16796*	Lemma 3 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- E = (ExtStrCat ` U )   &    |- B = ( Base ` E )   =>    |- ( ph -> F : C --> B )
Theorem	embedsetcestrclem 16797*	Lemma for embedsetcestrc 16807. (Contributed by AV, 31-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- E = (ExtStrCat ` U )   &    |- B = ( Base ` E )   =>    |- ( ph -> F : C -1-1-> B )
Theorem	funcsetcestrclem4 16798*	Lemma 4 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- ( ph -> G = ( x e. C , y e. C |-> ( _I |` ( y ^m x ) ) ) )   =>    |- ( ph -> G Fn ( C X. C ) )
Theorem	funcsetcestrclem5 16799*	Lemma 5 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- ( ph -> G = ( x e. C , y e. C |-> ( _I |` ( y ^m x ) ) ) )   =>    |- ( ( ph /\ ( X e. C /\ Y e. C ) ) -> ( X G Y ) = ( _I |` ( Y ^m X ) ) )
Theorem	funcsetcestrclem6 16800*	Lemma 6 for funcsetcestrc 16804. (Contributed by AV, 27-Mar-2020.)
|- S = ( SetCat ` U )   &    |- C = ( Base ` S )   &    |- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )   &    |- ( ph -> U e. WUni )   &    |- ( ph -> om e. U )   &    |- ( ph -> G = ( x e. C , y e. C |-> ( _I |` ( y ^m x ) ) ) )   =>    |- ( ( ph /\ ( X e. C /\ Y e. C ) /\ H e. ( Y ^m X ) ) -> ( ( X G Y ) ` H ) = H )