P. 164 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mrissmrid 16301	In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> S e. I )   &    |- ( ph -> T C_ S )   =>    |- ( ph -> T e. I )
Theorem	mreexd 16302*	In a Moore system, the closure operator is said to have the exchange property if, for all elements y and z of the base set and subsets S of the base set such that z is in the closure of ( S u. { y } ) but not in the closure of S, y is in the closure of ( S u. { z } ) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> X e. V )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S C_ X )   &    |- ( ph -> Y e. X )   &    |- ( ph -> Z e. ( N ` ( S u. { Y } ) ) )   &    |- ( ph -> -. Z e. ( N ` S ) )   =>    |- ( ph -> Y e. ( N ` ( S u. { Z } ) ) )
Theorem	mreexmrid 16303*	In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S e. I )   &    |- ( ph -> Y e. X )   &    |- ( ph -> -. Y e. ( N ` S ) )   =>    |- ( ph -> ( S u. { Y } ) e. I )
Theorem	mreexexlemd 16304*	This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 16308. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> X e. J )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F ~~ K \/ G ~~ K ) )   &    |- ( ph -> A. t A. u e. ~P ( X \ t ) A. v e. ~P ( X \ t ) ( ( ( u ~~ K \/ v ~~ K ) /\ u C_ ( N ` ( v u. t ) ) /\ ( u u. t ) e. I ) -> E. i e. ~P v ( u ~~ i /\ ( i u. t ) e. I ) ) )   =>    |- ( ph -> E. j e. ~P G ( F ~~ j /\ ( j u. H ) e. I ) )
Theorem	mreexexlem2d 16305*	Used in mreexexlem4d 16307 to prove the induction step in mreexexd 16308. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> Y e. F )   =>    |- ( ph -> E. g e. G ( -. g e. ( F \ { Y } ) /\ ( ( F \ { Y } ) u. ( H u. { g } ) ) e. I ) )
Theorem	mreexexlem3d 16306*	Base case of the induction in mreexexd 16308. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F = (/) \/ G = (/) ) )   =>    |- ( ph -> E. i e. ~P G ( F ~~ i /\ ( i u. H ) e. I ) )
Theorem	mreexexlem4d 16307*	Induction step of the induction in mreexexd 16308. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> L e. om )   &    |- ( ph -> A. h A. f e. ~P ( X \ h ) A. g e. ~P ( X \ h ) ( ( ( f ~~ L \/ g ~~ L ) /\ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) -> E. j e. ~P g ( f ~~ j /\ ( j u. h ) e. I ) ) )   &    |- ( ph -> ( F ~~ suc L \/ G ~~ suc L ) )   =>    |- ( ph -> E. j e. ~P G ( F ~~ j /\ ( j u. H ) e. I ) )
Theorem	mreexexd 16308*	Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if F and G are disjoint from H, ( F u. H ) is independent, F is contained in the closure of ( G u. H ), and either F or G is finite, then there is a subset q of G equinumerous to F such that ( q u. H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either ( A \ B ) or ( B \ A ) is finite. The theorem is proven by induction using mreexexlem3d 16306 for the base case and mreexexlem4d 16307 for the induction step. (Contributed by David Moews, 1-May-2017.) Removed dependencies on ax-rep 4771 and ax-ac2 9285. (Revised by Brendan Leahy, 2-Jun-2021.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F e. Fin \/ G e. Fin ) )   =>    |- ( ph -> E. q e. ~P G ( F ~~ q /\ ( q u. H ) e. I ) )
Theorem	mreexexdOLD 16309*	Obsolete proof of mreexexd 16308 as of 2-Jun-2021. Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if F and G are disjoint from H, ( F u. H ) is independent, F is contained in the closure of ( G u. H ), and either F or G is finite, then there is a subset q of G equinumerous to F such that ( q u. H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either ( A \ B ) or ( B \ A ) is finite. The theorem is proven by induction using mreexexlem3d 16306 for the base case and mreexexlem4d 16307 for the induction step. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F e. Fin \/ G e. Fin ) )   =>    |- ( ph -> E. q e. ~P G ( F ~~ q /\ ( q u. H ) e. I ) )
Theorem	mreexdomd 16310*	In a Moore system whose closure operator has the exchange property, if S is independent and contained in the closure of T, and either S or T is finite, then T dominates S. This is an immediate consequence of mreexexd 16308. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S C_ ( N ` T ) )   &    |- ( ph -> T C_ X )   &    |- ( ph -> ( S e. Fin \/ T e. Fin ) )   &    |- ( ph -> S e. I )   =>    |- ( ph -> S ~<_ T )
Theorem	mreexfidimd 16311*	In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 16310 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S e. I )   &    |- ( ph -> T e. I )   &    |- ( ph -> S e. Fin )   &    |- ( ph -> ( N ` S ) = ( N ` T ) )   =>    |- ( ph -> S ~~ T )
7.2.3  Algebraic closure systems
Theorem	isacs 16312*	A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
Theorem	acsmre 16313	Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- ( C e. (ACS ` X ) -> C e. (Moore ` X ) )
Theorem	isacs2 16314*	In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) ) )
Theorem	acsfiel 16315*	A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (ACS ` X ) -> ( S e. C <-> ( S C_ X /\ A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )
Theorem	acsfiel2 16316*	A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (ACS ` X ) /\ S C_ X ) -> ( S e. C <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) )
Theorem	acsmred 16317	An algebraic closure system is also a Moore system. Deduction form of acsmre 16313. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (ACS ` X ) )   =>    |- ( ph -> A e. (Moore ` X ) )
Theorem	isacs1i 16318*	A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (ACS ` X ) )
Theorem	mreacs 16319	Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X e. V -> (ACS ` X ) e. (Moore ` ~P X ) )
Theorem	acsfn 16320*	Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( ( X e. V /\ K e. X ) /\ ( T C_ X /\ T e. Fin ) ) -> { a e. ~P X | ( T C_ a -> K e. a ) } e. (ACS ` X ) )
Theorem	acsfn0 16321*	Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ K e. X ) -> { a e. ~P X | K e. a } e. (ACS ` X ) )
Theorem	acsfn1 16322*	Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. X E e. X ) -> { a e. ~P X | A. b e. a E e. a } e. (ACS ` X ) )
Theorem	acsfn1c 16323*	Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. K A. c e. X E e. X ) -> { a e. ~P X | A. b e. K A. c e. a E e. a } e. (ACS ` X ) )
Theorem	acsfn2 16324*	Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. X A. c e. X E e. X ) -> { a e. ~P X | A. b e. a A. c e. a E e. a } e. (ACS ` X ) )
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories
Syntax	ccat 16325	Extend class notation with the class of categories.
class Cat
Syntax	ccid 16326	Extend class notation with the identity arrow of a category.
class Id
Syntax	chomf 16327	Extend class notation to include functionalized Hom-set extractor.
class Hom f
Syntax	ccomf 16328	Extend class notation to include functionalized composition operation.
class compf
Definition	df-cat 16329*	A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ( ( Base ` c )), the morphisms "hom" ( ( Hom ` c )) and the composition law "o" ( (comp ` c )). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 16330. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
|- Cat = { c | [. ( Base ` c ) / b ]. [. ( Hom ` c ) / h ]. [. (comp ` c ) / o ]. A. x e. b ( E. g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) /\ A. y e. b A. z e. b A. f e. ( x h y ) A. g e. ( y h z ) ( ( g ( <. x , y >. o z ) f ) e. ( x h z ) /\ A. w e. b A. k e. ( z h w ) ( ( k ( <. y , z >. o w ) g ) ( <. x , y >. o w ) f ) = ( k ( <. x , z >. o w ) ( g ( <. x , y >. o z ) f ) ) ) ) }
Definition	df-cid 16330*	Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- Id = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ [_ (comp ` c ) / o ]_ ( x e. b |-> ( iota_ g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) ) ) )
Definition	df-homf 16331*	Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- Hom f = ( c e. _V |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( x ( Hom ` c ) y ) ) )
Definition	df-comf 16332*	Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- compf = ( c e. _V |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) , y e. ( Base ` c ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` c ) y ) , f e. ( ( Hom ` c ) ` x ) |-> ( g ( x (comp ` c ) y ) f ) ) ) )
Theorem	iscat 16333*	The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   =>    |- ( C e. V -> ( C e. Cat <-> A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) ) )
Theorem	iscatd 16334*	Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. V )   &    |- ( ( ph /\ x e. B ) -> .1. e. ( x H x ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( y H x ) ) ) -> ( .1. ( <. y , x >. .x. x ) f ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( x H y ) ) ) -> ( f ( <. x , x >. .x. y ) .1. ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) ) ) -> ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) )   &    |- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( z e. B /\ w e. B ) ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) /\ k e. ( z H w ) ) ) -> ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) )   =>    |- ( ph -> C e. Cat )
Theorem	catidex 16335*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> E. g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) )
Theorem	catideu 16336*	Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> E! g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) )
Theorem	cidfval 16337*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   =>    |- ( ph -> .1. = ( x e. B |-> ( iota_ g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) ) ) )
Theorem	cidval 16338*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .1. ` X ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )
Theorem	cidffn 16339	The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
|- Id Fn Cat
Theorem	cidfn 16340	The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- B = ( Base ` C )   &    |- .1. = ( Id ` C )   =>    |- ( C e. Cat -> .1. Fn B )
Theorem	catidd 16341*	Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. Cat )   &    |- ( ( ph /\ x e. B ) -> .1. e. ( x H x ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( y H x ) ) ) -> ( .1. ( <. y , x >. .x. x ) f ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( x H y ) ) ) -> ( f ( <. x , x >. .x. y ) .1. ) = f )   =>    |- ( ph -> ( Id ` C ) = ( x e. B |-> .1. ) )
Theorem	iscatd2 16342*	Version of iscatd 16334 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. V )   &    |- ( ps <-> ( ( x e. B /\ y e. B ) /\ ( z e. B /\ w e. B ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) /\ k e. ( z H w ) ) ) )   &    |- ( ( ph /\ y e. B ) -> .1. e. ( y H y ) )   &    |- ( ( ph /\ ps ) -> ( .1. ( <. x , y >. .x. y ) f ) = f )   &    |- ( ( ph /\ ps ) -> ( g ( <. y , y >. .x. z ) .1. ) = g )   &    |- ( ( ph /\ ps ) -> ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) )   &    |- ( ( ph /\ ps ) -> ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) )   =>    |- ( ph -> ( C e. Cat /\ ( Id ` C ) = ( y e. B |-> .1. ) ) )
Theorem	catidcl 16343	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .1. ` X ) e. ( X H X ) )
Theorem	catlid 16344	Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- .x. = (comp ` C )   &    |- ( ph -> Y e. B )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( ( .1. ` Y ) ( <. X , Y >. .x. Y ) F ) = F )
Theorem	catrid 16345	Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- .x. = (comp ` C )   &    |- ( ph -> Y e. B )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( F ( <. X , X >. .x. Y ) ( .1. ` X ) ) = F )
Theorem	catcocl 16346	Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) e. ( X H Z ) )
Theorem	catass 16347	Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   &    |- ( ph -> W e. B )   &    |- ( ph -> K e. ( Z H W ) )   =>    |- ( ph -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) )
Theorem	0catg 16348	Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- ( ( C e. V /\ (/) = ( Base ` C ) ) -> C e. Cat )
Theorem	0cat 16349	The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- (/) e. Cat
Theorem	homffval 16350*	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   =>    |- F = ( x e. B , y e. B |-> ( x H y ) )
Theorem	fnhomeqhomf 16351	If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   =>    |- ( H Fn ( B X. B ) -> F = H )
Theorem	homfval 16352	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X F Y ) = ( X H Y ) )
Theorem	homffn 16353	The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   =>    |- F Fn ( B X. B )
Theorem	homfeq 16354*	Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- H = ( Hom ` C )   &    |- J = ( Hom ` D )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> B = ( Base ` D ) )   =>    |- ( ph -> ( ( Hom f ` C ) = ( Hom f ` D ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) ) )
Theorem	homfeqd 16355	If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ( Base ` C ) = ( Base ` D ) )   &    |- ( ph -> ( Hom ` C ) = ( Hom ` D ) )   =>    |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
Theorem	homfeqbas 16356	Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   =>    |- ( ph -> ( Base ` C ) = ( Base ` D ) )
Theorem	homfeqval 16357	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- J = ( Hom ` D )   &    |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X H Y ) = ( X J Y ) )
Theorem	comfffval 16358*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   =>    |- O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) H y ) , f e. ( H ` x ) |-> ( g ( x .x. y ) f ) ) )
Theorem	comffval 16359*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( <. X , Y >. O Z ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) )
Theorem	comfval 16360	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. O Z ) F ) = ( G ( <. X , Y >. .x. Z ) F ) )
Theorem	comfffval2 16361*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom f ` C )   &    |- .x. = (comp ` C )   =>    |- O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) H y ) , f e. ( H ` x ) |-> ( g ( x .x. y ) f ) ) )
Theorem	comffval2 16362*	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom f ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( <. X , Y >. O Z ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. .x. Z ) f ) ) )
Theorem	comfval2 16363	Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom f ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. O Z ) F ) = ( G ( <. X , Y >. .x. Z ) F ) )
Theorem	comfffn 16364	The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   =>    |- O Fn ( ( B X. B ) X. B )
Theorem	comffn 16365	The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- O = (compf ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( <. X , Y >. O Z ) Fn ( ( Y H Z ) X. ( X H Y ) ) )
Theorem	comfeq 16366*	Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- .x. = (comp ` C )   &    |- .xb = (comp ` D )   &    |- H = ( Hom ` C )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> B = ( Base ` D ) )   &    |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   =>    |- ( ph -> ( (compf ` C ) = (compf ` D ) <-> A. x e. B A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( g ( <. x , y >. .x. z ) f ) = ( g ( <. x , y >. .xb z ) f ) ) )
Theorem	comfeqd 16367	Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> (comp ` C ) = (comp ` D ) )   &    |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   =>    |- ( ph -> (compf ` C ) = (compf ` D ) )
Theorem	comfeqval 16368	Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- .xb = (comp ` D )   &    |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> (compf ` C ) = (compf ` D ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G ( <. X , Y >. .xb Z ) F ) )
Theorem	catpropd 16369	Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> (compf ` C ) = (compf ` D ) )   &    |- ( ph -> C e. V )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( C e. Cat <-> D e. Cat ) )
Theorem	cidpropd 16370	Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> (compf ` C ) = (compf ` D ) )   &    |- ( ph -> C e. V )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( Id ` C ) = ( Id ` D ) )
8.1.2  Opposite category
Syntax	coppc 16371	The opposite category operation.
class oppCat
Definition	df-oppc 16372*	Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- oppCat = ( f e. _V |-> ( ( f sSet <. ( Hom ` ndx ) , tpos ( Hom ` f ) >. ) sSet <. (comp ` ndx ) , ( u e. ( ( Base ` f ) X. ( Base ` f ) ) , z e. ( Base ` f ) |-> tpos ( <. z , ( 2nd ` u ) >. (comp ` f ) ( 1st ` u ) ) ) >. ) )
Theorem	oppcval 16373*	Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- O = (oppCat ` C )   =>    |- ( C e. V -> O = ( ( C sSet <. ( Hom ` ndx ) , tpos H >. ) sSet <. (comp ` ndx ) , ( u e. ( B X. B ) , z e. B |-> tpos ( <. z , ( 2nd ` u ) >. .x. ( 1st ` u ) ) ) >. ) )
Theorem	oppchomfval 16374	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- H = ( Hom ` C )   &    |- O = (oppCat ` C )   =>    |- tpos H = ( Hom ` O )
Theorem	oppchom 16375	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- H = ( Hom ` C )   &    |- O = (oppCat ` C )   =>    |- ( X ( Hom ` O ) Y ) = ( Y H X )
Theorem	oppccofval 16376	Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- .x. = (comp ` C )   &    |- O = (oppCat ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( <. X , Y >. (comp ` O ) Z ) = tpos ( <. Z , Y >. .x. X ) )
Theorem	oppcco 16377	Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- .x. = (comp ` C )   &    |- O = (oppCat ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( G ( <. X , Y >. (comp ` O ) Z ) F ) = ( F ( <. Z , Y >. .x. X ) G ) )
Theorem	oppcbas 16378	Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- O = (oppCat ` C )   &    |- B = ( Base ` C )   =>    |- B = ( Base ` O )
Theorem	oppccatid 16379	Lemma for oppccat 16382. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   =>    |- ( C e. Cat -> ( O e. Cat /\ ( Id ` O ) = ( Id ` C ) ) )
Theorem	oppchomf 16380	Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
|- O = (oppCat ` C )   &    |- H = ( Hom f ` C )   =>    |- tpos H = ( Hom f ` O )
Theorem	oppcid 16381	Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- O = (oppCat ` C )   &    |- B = ( Id ` C )   =>    |- ( C e. Cat -> ( Id ` O ) = B )
Theorem	oppccat 16382	An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- O = (oppCat ` C )   =>    |- ( C e. Cat -> O e. Cat )
Theorem	2oppcbas 16383	The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 16397. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   &    |- B = ( Base ` C )   =>    |- B = ( Base ` (oppCat ` O ) )
Theorem	2oppchomf 16384	The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 16397. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   =>    |- ( Hom f ` C ) = ( Hom f ` (oppCat ` O ) )
Theorem	2oppccomf 16385	The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16397. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   =>    |- (compf ` C ) = (compf ` (oppCat ` O ) )
Theorem	oppchomfpropd 16386	If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   =>    |- ( ph -> ( Hom f ` (oppCat ` C ) ) = ( Hom f ` (oppCat ` D ) ) )
Theorem	oppccomfpropd 16387	If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> (compf ` C ) = (compf ` D ) )   =>    |- ( ph -> (compf ` (oppCat ` C ) ) = (compf ` (oppCat ` D ) ) )
8.1.3  Monomorphisms and epimorphisms
Syntax	cmon 16388	Extend class notation with the class of all monomorphisms.
class Mono
Syntax	cepi 16389	Extend class notation with the class of all epimorphisms.
class Epi
Definition	df-mon 16390*	Function returning the monomorphisms of the category c. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
|- Mono = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) )
Definition	df-epi 16391	Function returning the epimorphisms of the category c. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
|- Epi = ( c e. Cat |-> tpos (Mono ` (oppCat ` c ) ) )
Theorem	monfval 16392*	Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- M = (Mono ` C )   &    |- ( ph -> C e. Cat )   =>    |- ( ph -> M = ( x e. B , y e. B |-> { f e. ( x H y ) | A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) } ) )
Theorem	ismon 16393*	Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- M = (Mono ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X H Y ) /\ A. z e. B Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) ) )
Theorem	ismon2 16394*	Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- M = (Mono ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X H Y ) /\ A. z e. B A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) ) ) )
Theorem	monhom 16395	A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- M = (Mono ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X M Y ) C_ ( X H Y ) )
Theorem	moni 16396	Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- M = (Mono ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X M Y ) )   &    |- ( ph -> G e. ( Z H X ) )   &    |- ( ph -> K e. ( Z H X ) )   =>    |- ( ph -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) <-> G = K ) )
Theorem	monpropd 16397	If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )   &    |- ( ph -> (compf ` C ) = (compf ` D ) )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> D e. Cat )   =>    |- ( ph -> (Mono ` C ) = (Mono ` D ) )
Theorem	oppcmon 16398	A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   &    |- ( ph -> C e. Cat )   &    |- M = (Mono ` O )   &    |- E = (Epi ` C )   =>    |- ( ph -> ( X M Y ) = ( Y E X ) )
Theorem	oppcepi 16399	An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- O = (oppCat ` C )   &    |- ( ph -> C e. Cat )   &    |- E = (Epi ` O )   &    |- M = (Mono ` C )   =>    |- ( ph -> ( X E Y ) = ( Y M X ) )
Theorem	isepi 16400*	Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- E = (Epi ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( F e. ( X E Y ) <-> ( F e. ( X H Y ) /\ A. z e. B Fun `' ( g e. ( Y H z ) |-> ( g ( <. X , Y >. .x. z ) F ) ) ) ) )