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Theorem List for Metamath Proof Explorer - 17101-17200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcatchomfval 17101* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))

Theoremcatchom 17102 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌))

Theoremcatccofval 17103* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))

Theoremcatcco 17104 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 Func 𝑌))    &   (𝜑𝐺 ∈ (𝑌 Func 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺func 𝐹))

Theoremcatccatid 17105* Lemma for catccat 17107. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))

Theoremcatcid 17106 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 = (idfunc𝑋)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) = 𝐼)

Theoremcatccat 17107 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 "𝑈 -large".) (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremresscatc 17108 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)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Theoremcatcisolem 17109* Lemma for catciso 17110. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Inv‘𝐶)    &   𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))    &   (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)    &   (𝜑𝐹:𝑅1-1-onto𝑆)       (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)

Theoremcatciso 17110 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)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝐹):𝑅1-1-onto𝑆)))

Theoremcatcoppccl 17111 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑋𝐵)       (𝜑𝑂𝐵)

Theoremcatcfuccl 17112 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑𝑄𝐵)

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 17116.

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 17118. 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 17114 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 17119, whereas the composition is the ordinary composition of functions, see estrccofval 17122 and estrcco 17123.

It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 17126 with the identity function as identity arrow, see estrcid 17127.

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 42820 and dfringc2 42866, but with special compositions). With this definitions, however, theorem rngcresringcat 42878 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 17079, 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 17116. The claimed equivalence is proven by equivestrcsetc 17146. 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 42807 and df-ringc 42853.

In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 42878, 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 17146. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 17161.

Remark: equivestrcsetc 17146 as well as embedsetcestrc 17161 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: ω ∈ 𝑈 (see wunndx 16244), but it would be currently sufficient to assume that 1 ∈ 𝑈, because the index value of the base set extractor is hard-coded as 1, see basendx 16287.

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 ∈ Ring → 𝐴 ∈ Grp is valid (see ringgrp 18907) 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 ∈ Grp → 𝑔 Struct ⟨(Base‘ndx), (+g‘ndx)⟩

𝑟 ∈ Ring → 𝑟 Struct ⟨(Base‘ndx), (+g‘ndx), (.r‘ndx)⟩

Theoremfncnvimaeqv 17113 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)

Theorembascnvimaeqv 17114 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

Syntaxcestrc 17115 Extend class notation to include the category ExtStr.
class ExtStrCat

Definitiondf-estrc 17116* 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 17114 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 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})

Theoremestrcval 17117* Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))    &   (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremestrcbas 17118 Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)       (𝜑𝑈 = (Base‘𝐶))

Theoremestrchomfval 17119* Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremestrchom 17120 The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)       (𝜑 → (𝑋𝐻𝑌) = (𝐵𝑚 𝐴))

Theoremelestrchom 17121 A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝐴𝐵))

Theoremestrccofval 17122* Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))

Theoremestrcco 17123 Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   𝐴 = (Base‘𝑋)    &   𝐵 = (Base‘𝑌)    &   𝐷 = (Base‘𝑍)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐷)       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremestrcbasbas 17124 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)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝐸𝐵) → (Base‘𝐸) ∈ 𝑈)

Theoremestrccatid 17125* Lemma for estrccat 17126. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ (Base‘𝑥)))))

Theoremestrccat 17126 The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremestrcid 17127 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)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)       (𝜑 → ( 1𝑋) = ( I ↾ (Base‘𝑋)))

Theoremestrchomfn 17128 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)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 Fn (𝑈 × 𝑈))

Theoremestrchomfeqhom 17129 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)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (Homf𝐶) = 𝐻)

Theoremestrreslem1 17130 Lemma 1 for estrres 17133. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)       (𝜑𝐵 = (Base‘𝐶))

Theoremestrreslem2 17131 Lemma 2 for estrres 17133. (Contributed by AV, 14-Mar-2020.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)       (𝜑 → (Base‘ndx) ∈ dom 𝐶)

TheoremestrresOLD 17132 Obsolete version of estrres 17133 as of 3-Jul-2022. (Contributed by AV, 15-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)    &   (𝜑𝐴𝑈)    &   (𝜑𝐺𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐶s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})

Theoremestrres 17133 Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.) (Revised by AV, 3-Jul-2022.)
(𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐻𝑋)    &   (𝜑·𝑌)    &   (𝜑𝐺𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝐶s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})

Theoremfuncestrcsetclem1 17134* Lemma 1 for funcestrcsetc 17143. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))

Theoremfuncestrcsetclem2 17135* Lemma 2 for funcestrcsetc 17143. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)

Theoremfuncestrcsetclem3 17136* Lemma 3 for funcestrcsetc 17143. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)

Theoremfuncestrcsetclem4 17137* Lemma 4 for funcestrcsetc 17143. (Contributed by AV, 22-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐺 Fn (𝐵 × 𝐵))

Theoremfuncestrcsetclem5 17138* Lemma 5 for funcestrcsetc 17143. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   𝑀 = (Base‘𝑋)    &   𝑁 = (Base‘𝑌)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁𝑚 𝑀)))

Theoremfuncestrcsetclem6 17139* Lemma 6 for funcestrcsetc 17143. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   𝑀 = (Base‘𝑋)    &   𝑁 = (Base‘𝑌)       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑁𝑚 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Theoremfuncestrcsetclem7 17140* Lemma 7 for funcestrcsetc 17143. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))

Theoremfuncestrcsetclem8 17141* Lemma 8 for funcestrcsetc 17143. (Contributed by AV, 15-Feb-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))

Theoremfuncestrcsetclem9 17142* Lemma 9 for funcestrcsetc 17143. (Contributed by AV, 23-Mar-2020.)
𝐸 = (ExtStrCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝐸)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

Theoremfuncestrcsetc 17143* 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Func 𝑆)𝐺)

Theoremfthestrcsetc 17144* 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Faith 𝑆)𝐺)

Theoremfullestrcsetc 17145* 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))       (𝜑𝐹(𝐸 Full 𝑆)𝐺)

Theoremequivestrcsetc 17146* 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))    &   (𝜑 → (Base‘ndx) ∈ 𝑈)       (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏𝐶𝑎𝐵𝑖 𝑖:𝑏1-1-onto→(𝐹𝑎)))

Theoremsetc1strwun 17147 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ 𝑈)

Theoremfuncsetcestrclem1 17148* Lemma 1 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))       ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})

Theoremfuncsetcestrclem2 17149* Lemma 2 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)       ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ 𝑈)

Theoremfuncsetcestrclem3 17150* Lemma 3 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑𝐹:𝐶𝐵)

Theoremembedsetcestrclem 17151* Lemma for embedsetcestrc 17161. (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑𝐹:𝐶1-1𝐵)

Theoremfuncsetcestrclem4 17152* Lemma 4 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       (𝜑𝐺 Fn (𝐶 × 𝐶))

Theoremfuncsetcestrclem5 17153* Lemma 5 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌𝑚 𝑋)))

Theoremfuncsetcestrclem6 17154* Lemma 6 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))       ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌𝑚 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Theoremfuncsetcestrclem7 17155* Lemma 7 for funcsetcestrc 17158. (Contributed by AV, 27-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑𝑋𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹𝑋)))

Theoremfuncsetcestrclem8 17156* Lemma 8 for funcsetcestrc 17158. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑆)𝑌)⟶((𝐹𝑋)(Hom ‘𝐸)(𝐹𝑌)))

Theoremfuncsetcestrclem9 17157* Lemma 9 for funcsetcestrc 17158. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

Theoremfuncsetcestrc 17158* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Func 𝐸)𝐺)

Theoremfthsetcestrc 17159* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is faithful. (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Faith 𝐸)𝐺)

Theoremfullsetcestrc 17160* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)       (𝜑𝐹(𝑆 Full 𝐸)𝐺)

Theoremembedsetcestrc 17161* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is an embedding. According to definition 3.27 (1) of [Adamek] p. 34, a functor "F is called an embedding provided that F is injective on morphisms", or according to remark 3.28 (1) in [Adamek] p. 34, "a functor is an embedding if and only if it is faithful and injective on objects". (Contributed by AV, 31-Mar-2020.)
𝑆 = (SetCat‘𝑈)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝐸)       (𝜑 → (𝐹(𝑆 Faith 𝐸)𝐺𝐹:𝐶1-1𝐵))

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 17162 Extend class notation with the product of two categories.
class ×c

Syntaxc1stf 17163 Extend class notation with the first projection functor.
class 1stF

Syntaxc2ndf 17164 Extend class notation with the second projection functor.
class 2ndF

Syntaxcprf 17165 Extend class notation with the functor pairing operation.
class ⟨,⟩F

Definitiondf-xpc 17166* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏(𝑢𝑏, 𝑣𝑏 ↦ (((1st𝑢)(Hom ‘𝑟)(1st𝑣)) × ((2nd𝑢)(Hom ‘𝑠)(2nd𝑣)))) / {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦𝑏 ↦ (𝑔 ∈ ((2nd𝑥)𝑦), 𝑓 ∈ (𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩(comp‘𝑟)(1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩(comp‘𝑠)(2nd𝑦))(2nd𝑓))⟩))⟩})

Definitiondf-1stf 17167* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(1st𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)

Definitiondf-2ndf 17168* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ((Base‘𝑟) × (Base‘𝑠)) / 𝑏⟨(2nd𝑏), (𝑥𝑏, 𝑦𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)

Definitiondf-prf 17169* Define the pairing operation for functors (which takes two functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐸 and produces (𝐹 ⟨,⟩F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩)

Theoremfnxpc 17170 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
×c Fn (V × V)

Theoremxpcval 17171* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐵 = (𝑋 × 𝑌))    &   (𝜑𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))    &   (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))       (𝜑𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Theoremxpcbas 17172 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)       (𝑋 × 𝑌) = (Base‘𝑇)

Theoremxpchomfval 17173* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐾 = (Hom ‘𝑇)       𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))

Theoremxpchom 17174 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐾 = (Hom ‘𝑇)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))

Theoremrelxpchom 17175 A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐾 = (Hom ‘𝑇)       Rel (𝑋𝐾𝑌)

Theoremxpccofval 17176* Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))

Theoremxpcco 17177 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩)

Theoremxpcco1st 17178 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &   𝑂 = (comp‘𝑇)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))    &    · = (comp‘𝐶)       (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))

Theoremxpcco2nd 17179 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &   𝑂 = (comp‘𝑇)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))    &    · = (comp‘𝐷)       (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)))

Theoremxpchom2 17180 Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝑀𝑋)    &   (𝜑𝑁𝑌)    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑌)    &   𝐾 = (Hom ‘𝑇)       (𝜑 → (⟨𝑀, 𝑁𝐾𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))

Theoremxpcco2 17181 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝑀𝑋)    &   (𝜑𝑁𝑌)    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑌)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑𝐹 ∈ (𝑀𝐻𝑃))    &   (𝜑𝐺 ∈ (𝑁𝐽𝑄))    &   (𝜑𝐾 ∈ (𝑃𝐻𝑅))    &   (𝜑𝐿 ∈ (𝑄𝐽𝑆))       (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃· 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄 𝑆)𝐺)⟩)

Theoremxpccatid 17182* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐼 = (Id‘𝐶)    &   𝐽 = (Id‘𝐷)       (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)))

Theoremxpcid 17183 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐼 = (Id‘𝐶)    &   𝐽 = (Id‘𝐷)    &    1 = (Id‘𝑇)    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)       (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼𝑅), (𝐽𝑆)⟩)

Theoremxpccat 17184 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝑇 ∈ Cat)

Theorem1stfval 17185* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)       (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)

Theorem1stf1 17186 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)    &   (𝜑𝑅𝐵)       (𝜑 → ((1st𝑃)‘𝑅) = (1st𝑅))

Theorem1stf2 17187 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)    &   (𝜑𝑅𝐵)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))

Theorem2ndfval 17188* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)       (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)

Theorem2ndf1 17189 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)    &   (𝜑𝑅𝐵)       (𝜑 → ((1st𝑄)‘𝑅) = (2nd𝑅))

Theorem2ndf2 17190 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)    &   (𝜑𝑅𝐵)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Theorem1stfcl 17191 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)       (𝜑𝑃 ∈ (𝑇 Func 𝐶))

Theorem2ndfcl 17192 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)       (𝜑𝑄 ∈ (𝑇 Func 𝐷))

Theoremprfval 17193* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)

Theoremprf1 17194 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝑃)‘𝑋) = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)

Theoremprf2fval 17195* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))

Theoremprf2 17196 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩)

Theoremprfcl 17197 The pairing of functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐷 is a functor 𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝑇 = (𝐷 ×c 𝐸)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑𝑃 ∈ (𝐶 Func 𝑇))

Theoremprf1st 17198 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

Theoremprf2nd 17199 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)

Theorem1st2ndprf 17200 Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑇 = (𝐷 ×c 𝐸)    &   (𝜑𝐹 ∈ (𝐶 Func 𝑇))    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)       (𝜑𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))

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