P. 184 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	setc2obas 18301	∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 18302 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17874 and cat1 18304. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ 𝐶 = (SetCat‘2o)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅)
Theorem	setc2ohom 18302	(SetCat‘2o) is a category (provable from setccat 18292 and 2oex 8519) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 18301. Notably, the empty set ∅ is simultaneously an object (setc2obas 18301), an identity morphism from ∅ to ∅ (setcid 18293 or thincid 50782), and a non-identity morphism from ∅ to 1o. See cat1lem 18303 and cat1 18304 for a more general statement. This category is also thin (setc2othin 50816), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 50814 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ 𝐶 = (SetCat‘2o)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o))
Theorem	cat1lem 18303*	The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 18304. (Contributed by Zhi Wang, 15-Sep-2024.)
⊢ 𝐶 = (SetCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → ∅ ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → ∅ ≠ 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
Theorem	cat1 18304*	The definition of category df-cat 17874 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 18301 and setc2ohom 18302 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 18233 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
⊢ ∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
8.3.2  The category of categories
Syntax	ccatc 18305	Extend class notation to include the category Cat.
class CatCat
Definition	df-catc 18306*	Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e., "𝑢-small categories", see Definition 3.44. of [Adamek] p. 39), with functors as the morphisms (catchom 18310, elcatchom 50747). Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "𝑢-large categories", which can be expressed as (Cat ∖ 𝑢). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
Theorem	catcval 18307*	Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	catcbas 18308	Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
Theorem	catchomfval 18309*	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
Theorem	catchom 18310	Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌))
Theorem	catccofval 18311*	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
Theorem	catcco 18312	Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌 Func 𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘func 𝐹))
Theorem	catccatid 18313*	Lemma for catccat 18315. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ (idfunc‘𝑥))))
Theorem	catcid 18314	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = 𝐼)
Theorem	catccat 18315	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)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	resscatc 18316	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑈)    ⇒   ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷)))
Theorem	catcisolem 18317*	Lemma for catciso 18318. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Inv‘𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ◡((◡𝐹‘𝑥)𝐺(◡𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)    &   ⊢ (𝜑 → 𝐹:𝑅⟶1-1onto𝑆)    ⇒   ⊢ (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨◡𝐹, 𝐻⟩)
Theorem	catciso 18318	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. Note that "catciso.u" is redundant thanks to elbasfv 17425. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅⟶1-1onto𝑆)))
Theorem	catcbascl 18319	An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18324. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑈)
Theorem	catcslotelcl 18320	A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18324. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝑋) ∈ 𝑈)
Theorem	catcbaselcl 18321	The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18324. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
Theorem	catchomcl 18322	The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18324. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
Theorem	catcccocl 18323	The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 18324. (Contributed by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
Theorem	catcoppccl 18324	The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑂 = (oppCat‘𝑋)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝐵)
Theorem	catcfuccl 18325	The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑄 = (𝑋 FuncCat 𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑄 ∈ 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 18329. 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 18331. 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 18327 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 18332, whereas the composition is the ordinary composition of functions, see estrccofval 18335 and estrcco 18336. It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 18339 with the identity function as identity arrow, see estrcid 18340. 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 18340): At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordered pairs, see dfrngc2 20913 and dfringc2 20942, but with special compositions). With this definitions, however, Theorem rngcresringcat 20954 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 18283, 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 18329. The claimed equivalence is proven by equivestrcsetc 18358. 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 20902 and df-ringc 20931. In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 20954, 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 18358. Furthermore, it can be shown that SetCat is embedded into ExtStrCat, see embedsetcestrc 18373. Remark: equivestrcsetc 18358 as well as embedsetcestrc 18373 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 17405), 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 17428. 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) ⊆ Ob(B) and MorA(X,Y) ⊆ 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) ⊆ 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 20497) 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)⟩
Theorem	fncnvimaeqv 18326	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 18327	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 18328	Extend class notation to include the category ExtStr.
class ExtStrCat
Definition	df-estrc 18329*	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 18327 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), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	estrcval 18330*	Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	estrcbas 18331	Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑈 = (Base‘𝐶))
Theorem	estrchomfval 18332*	Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	estrchom 18333	The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝐵 ↑m 𝐴))
Theorem	elestrchom 18334	A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝐴⟶𝐵))
Theorem	estrccofval 18335*	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
Theorem	estrcco 18336	Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ 𝐴 = (Base‘𝑋)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐷 = (Base‘𝑍)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	estrcbasbas 18337	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    ⇒   ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → (Base‘𝐸) ∈ 𝑈)
Theorem	estrccatid 18338*	Lemma for estrccat 18339. (Contributed by AV, 8-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ (Base‘𝑥)))))
Theorem	estrccat 18339	The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	estrcid 18340	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ (Base‘𝑋)))
Theorem	estrchomfn 18341	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝑈 × 𝑈))
Theorem	estrchomfeqhom 18342	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (Homf ‘𝐶) = 𝐻)
Theorem	estrreslem1 18343	Lemma 1 for estrres 18345. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	estrreslem2 18344	Lemma 2 for estrres 18345. (Contributed by AV, 14-Mar-2020.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → · ∈ 𝑌)    ⇒   ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐶)
Theorem	estrres 18345	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.)
⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → · ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐶 ↾s 𝐴) sSet ⟨(Hom ‘ndx), 𝐺⟩) = {⟨(Base‘ndx), 𝐴⟩, ⟨(Hom ‘ndx), 𝐺⟩, ⟨(comp‘ndx), · ⟩})
Theorem	funcestrcsetclem1 18346*	Lemma 1 for funcestrcsetc 18355. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
Theorem	funcestrcsetclem2 18347*	Lemma 2 for funcestrcsetc 18355. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcestrcsetclem3 18348*	Lemma 3 for funcestrcsetc 18355. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcestrcsetclem4 18349*	Lemma 4 for funcestrcsetc 18355. (Contributed by AV, 22-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcestrcsetclem5 18350*	Lemma 5 for funcestrcsetc 18355. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ 𝑀 = (Base‘𝑋)    &   ⊢ 𝑁 = (Base‘𝑌)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀)))
Theorem	funcestrcsetclem6 18351*	Lemma 6 for funcestrcsetc 18355. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ 𝑀 = (Base‘𝑋)    &   ⊢ 𝑁 = (Base‘𝑌)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcestrcsetclem7 18352*	Lemma 7 for funcestrcsetc 18355. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	funcestrcsetclem8 18353*	Lemma 8 for funcestrcsetc 18355. (Contributed by AV, 15-Feb-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))
Theorem	funcestrcsetclem9 18354*	Lemma 9 for funcestrcsetc 18355. (Contributed by AV, 23-Mar-2020.)
⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcestrcsetc 18355*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺)
Theorem	fthestrcsetc 18356*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺)
Theorem	fullestrcsetc 18357*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    ⇒   ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺)
Theorem	equivestrcsetc 18358*	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))    &   ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏⟶1-1onto(𝐹‘𝑎)))
Theorem	setc1strwun 18359	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)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ 𝑈)
Theorem	funcsetcestrclem1 18360*	Lemma 1 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {⟨(Base‘ndx), 𝑋⟩})
Theorem	funcsetcestrclem2 18361*	Lemma 2 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcsetcestrclem3 18362*	Lemma 3 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐶⟶𝐵)
Theorem	embedsetcestrclem 18363*	Lemma for embedsetcestrc 18373. (Contributed by AV, 31-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐶⟶1-1𝐵)
Theorem	funcsetcestrclem4 18364*	Lemma 4 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶))
Theorem	funcsetcestrclem5 18365*	Lemma 5 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑m 𝑋)))
Theorem	funcsetcestrclem6 18366*	Lemma 6 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcsetcestrclem7 18367*	Lemma 7 for funcsetcestrc 18370. (Contributed by AV, 27-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
Theorem	funcsetcestrclem8 18368*	Lemma 8 for funcsetcestrc 18370. (Contributed by AV, 28-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑆)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌)))
Theorem	funcsetcestrclem9 18369*	Lemma 9 for funcsetcestrc 18370. (Contributed by AV, 28-Mar-2020.)
⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcsetcestrc 18370*	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)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺)
Theorem	fthsetcestrc 18371*	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)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Faith 𝐸)𝐺)
Theorem	fullsetcestrc 18372*	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)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺)
Theorem	embedsetcestrc 18373*	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)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → ω ∈ 𝑈)    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥))))    &   ⊢ 𝐸 = (ExtStrCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → (𝐹(𝑆 Faith 𝐸)𝐺 ∧ 𝐹:𝐶⟶1-1𝐵))
8.4  Categorical constructions
8.4.1  Product of categories
Syntax	cxpc 18374	Extend class notation with the product of two categories.
class ×c
Syntax	c1stf 18375	Extend class notation with the first projection functor.
class 1stF
Syntax	c2ndf 18376	Extend class notation with the second projection functor.
class 2ndF
Syntax	cprf 18377	Extend class notation with the functor pairing operation.
class ⟨,⟩F
Definition	df-xpc 18378*	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⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
Definition	df-1stf 18379*	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⦌⟨(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
Definition	df-2ndf 18380*	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⦌⟨(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))⟩)
Definition	df-prf 18381*	Define the pairing operation for functors (which takes two functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐸 and produces (𝐹 ⟨,⟩F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ ⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌⟨(𝑥 ∈ 𝑏 ↦ ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ ⟨((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)⟩))⟩)
Theorem	fnxpc 18382	The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
⊢ ×c Fn (V × V)
Theorem	xpcval 18383*	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)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌))    &   ⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))    &   ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))    ⇒   ⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
Theorem	xpcbas 18384	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)
Theorem	xpchomfval 18385*	Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
Theorem	xpchom 18386	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)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))
Theorem	relxpchom 18387	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)
Theorem	xpccofval 18388*	Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝑇)    ⇒   ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩))
Theorem	xpcco 18389	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)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ ∙ (2nd ‘𝑍))(2nd ‘𝐹))⟩)
Theorem	xpcco1st 18390	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)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩ · (1st ‘𝑍))(1st ‘𝐹)))
Theorem	xpcco2nd 18391	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)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))    &   ⊢ · = (comp‘𝐷)    ⇒   ⊢ (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)))
Theorem	xpchom2 18392	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)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑌)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))
Theorem	xpcco2 18393	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)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑌)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩ · 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩ ∙ 𝑆)𝐺)⟩)
Theorem	xpccatid 18394*	The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ 𝐽 = (Id‘𝐷)    ⇒   ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐼‘𝑥), (𝐽‘𝑦)⟩)))
Theorem	xpcid 18395	The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐶)    &   ⊢ 𝐽 = (Id‘𝐷)    &   ⊢ 1 = (Id‘𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    ⇒   ⊢ (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼‘𝑅), (𝐽‘𝑆)⟩)
Theorem	xpccat 18396	The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑇 ∈ Cat)
Theorem	1stfval 18397*	Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    ⇒   ⊢ (𝜑 → 𝑃 = ⟨(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
Theorem	1stf1 18398	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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = (1st ‘𝑅))
Theorem	1stf2 18399	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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑃 = (𝐶 1stF 𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))
Theorem	2ndfval 18400*	Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐻 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 2ndF 𝐷)    ⇒   ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)