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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremacsfiel 13401* 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.)
mrCls       ACS

Theoremacsfiel2 13402* 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.)
mrCls       ACS

Theoremisacs1i 13403* A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremmreacs 13404 Algebraicity is a composible property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS Moore

Theoremacsfn 13405* Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn0 13406* Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1 13407* Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn1c 13408* Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

Theoremacsfn2 13409* Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
ACS

PART 8  BASIC CATEGORY THEORY

8.1  Categories

8.1.1  Categories

Syntaxccat 13410 Extend class notation with the class of categories.

Syntaxccid 13411 Extend class notation with the identity arrow of a category.

Syntaxchomf 13412 Extend class notation to include functionalized Hom-set extractor.
f

Syntaxccomf 13413 Extend class notation to include functionalized composition operation.
compf

Definitiondf-cat 13414* 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 to 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. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
comp

Definitiondf-cid 13415* 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.)
comp

Definitiondf-homf 13416* Define the functionalized Hom-set operator, which is exactly like but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Definitiondf-comf 13417* 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 comp

Theoremiscat 13418* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremiscatd 13419* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatidex 13420* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatideu 13421* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidfval 13422* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremcidval 13423* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidffn 13424 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)

Theoremcidfn 13425 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcatidd 13426* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremiscatd2 13427* Version of iscatd2 13427 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp

Theoremcatidcl 13428 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcatlid 13429 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatrid 13430 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatcocl 13431 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatass 13432 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theorem0catg 13433 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theorem0cat 13434 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremproplem2 13435* Lemma for mndpropd 14233. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem 13436* Lemma for mndpropd 14233. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem3 13437 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremhomffval 13438* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfval 13439 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomffn 13440 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfeq 13441* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
f f

Theoremhomfeqd 13442 If two structures have the same slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqbas 13443 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqval 13444 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremcomfffval 13445* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomffval 13446* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfval 13447 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfffval2 13448* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomffval2 13449* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfval2 13450 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfffn 13451 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomffn 13452 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomfeq 13453* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp                            f f        compf compf

Theoremcomfeqd 13454 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp comp       f f        compf compf

Theoremcomfeqval 13455 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp       f f        compf compf

Theoremcatpropd 13456 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
f f        compf compf

Theoremcidpropd 13457 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf

8.1.2  Opposite category

Syntaxcoppc 13458 The opposite category operation.
oppCat

Definitiondf-oppc 13459* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat sSet tpos sSet comp tpos comp

Theoremoppcval 13460* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat       sSet tpos sSet comp tpos

Theoremoppchomfval 13461 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat       tpos

Theoremoppchom 13462 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccofval 13463 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp tpos

Theoremoppcco 13464 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp

Theoremoppcbas 13465 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccatid 13466 Lemma for oppccat 13469. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremoppchomf 13467 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       f        tpos f

Theoremoppcid 13468 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccat 13469 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theorem2oppcbas 13470 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              oppCat

Theorem2oppchomf 13471 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       f f oppCat

Theorem2oppccomf 13472 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13484. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       compf compfoppCat

Theoremoppchomfpropd 13473 If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        f oppCat f oppCat

Theoremoppccomfpropd 13474 If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       compfoppCat compfoppCat

8.1.3  Monomorphisms and epimorphisms

Syntaxcmon 13475 Extend class notation with the class of all monomorphisms.
Mono

Syntaxcepi 13476 Extend class notation with the class of all epimorphisms.
Epi

Definitiondf-mon 13477* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Mono comp

Definitiondf-epi 13478 Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Epi tpos MonooppCat

Theoremmonfval 13479* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Mono

Theoremismon 13480* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremismon2 13481* Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonhom 13482 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmoni 13483 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonpropd 13484 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.)
f f        compf compf                     Mono Mono

Theoremoppcmon 13485 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Mono       Epi

Theoremoppcepi 13486 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Epi       Mono

Theoremisepi 13487* Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremisepi2 13488* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepihom 13489 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepii 13490 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Epi

8.1.4  Sections, inverses, isomorphisms

Syntaxcsect 13491 Extend class notation with the sections of a morphism.
Sect

Syntaxcinv 13492 Extend class notation with the inverses of a morphism.
Inv

Syntaxciso 13493 Extend class notation with the class of all isomorphisms.

Definitiondf-sect 13494* Function returning the section relation in a category. Given arrows and , we say Sect, that is, is a section of , if . (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect comp

Definitiondf-inv 13495* The inverse relation in a category. Given arrows and , we say Inv, that is, is an inverse of , if is a section of and is a section of . (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv Sect Sect

Definitiondf-iso 13496* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv

Theoremsectffval 13497* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectfval 13498* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectss 13499 The section relation is a relation between morphisms from to and morphisms from to . (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect 13500 The property " is a section of ". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

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