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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisnat2 17901* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   π΅ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   π½ = (Hom β€˜π·)    &    Β· = (compβ€˜π·)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    &   (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))    β‡’   (πœ‘ β†’ (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ Xπ‘₯ ∈ 𝐡 (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€β„Ž ∈ (π‘₯𝐻𝑦)((π΄β€˜π‘¦)(⟨((1st β€˜πΉ)β€˜π‘₯), ((1st β€˜πΉ)β€˜π‘¦)⟩ Β· ((1st β€˜πΊ)β€˜π‘¦))((π‘₯(2nd β€˜πΉ)𝑦)β€˜β„Ž)) = (((π‘₯(2nd β€˜πΊ)𝑦)β€˜β„Ž)(⟨((1st β€˜πΉ)β€˜π‘₯), ((1st β€˜πΊ)β€˜π‘₯)⟩ Β· ((1st β€˜πΊ)β€˜π‘¦))(π΄β€˜π‘₯)))))
 
Theoremnatffn 17902 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    β‡’   π‘ Fn ((𝐢 Func 𝐷) Γ— (𝐢 Func 𝐷))
 
Theoremnatrcl 17903 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    β‡’   (𝐴 ∈ (𝐹𝑁𝐺) β†’ (𝐹 ∈ (𝐢 Func 𝐷) ∧ 𝐺 ∈ (𝐢 Func 𝐷)))
 
Theoremnat1st2nd 17904 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ (𝐹𝑁𝐺))    β‡’   (πœ‘ β†’ 𝐴 ∈ (⟨(1st β€˜πΉ), (2nd β€˜πΉ)βŸ©π‘βŸ¨(1st β€˜πΊ), (2nd β€˜πΊ)⟩))
 
Theoremnatixp 17905* A natural transformation is a function from the objects of 𝐢 to homomorphisms from 𝐹(π‘₯) to 𝐺(π‘₯). (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ (⟨𝐹, πΊβŸ©π‘βŸ¨πΎ, 𝐿⟩))    &   π΅ = (Baseβ€˜πΆ)    &   π½ = (Hom β€˜π·)    β‡’   (πœ‘ β†’ 𝐴 ∈ Xπ‘₯ ∈ 𝐡 ((πΉβ€˜π‘₯)𝐽(πΎβ€˜π‘₯)))
 
Theoremnatcl 17906 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ (⟨𝐹, πΊβŸ©π‘βŸ¨πΎ, 𝐿⟩))    &   π΅ = (Baseβ€˜πΆ)    &   π½ = (Hom β€˜π·)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π΄β€˜π‘‹) ∈ ((πΉβ€˜π‘‹)𝐽(πΎβ€˜π‘‹)))
 
Theoremnatfn 17907 A natural transformation is a function on the objects of 𝐢. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ (⟨𝐹, πΊβŸ©π‘βŸ¨πΎ, 𝐿⟩))    &   π΅ = (Baseβ€˜πΆ)    β‡’   (πœ‘ β†’ 𝐴 Fn 𝐡)
 
Theoremnati 17908 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ (⟨𝐹, πΊβŸ©π‘βŸ¨πΎ, 𝐿⟩))    &   π΅ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &    Β· = (compβ€˜π·)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑅 ∈ (π‘‹π»π‘Œ))    β‡’   (πœ‘ β†’ ((π΄β€˜π‘Œ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩ Β· (πΎβ€˜π‘Œ))((π‘‹πΊπ‘Œ)β€˜π‘…)) = (((π‘‹πΏπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘‹), (πΎβ€˜π‘‹)⟩ Β· (πΎβ€˜π‘Œ))(π΄β€˜π‘‹)))
 
Theoremwunnat 17909 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
(πœ‘ β†’ π‘ˆ ∈ WUni)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝐢 Nat 𝐷) ∈ π‘ˆ)
 
TheoremwunnatOLD 17910 Obsolete proof of wunnat 17909 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ WUni)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝐢 Nat 𝐷) ∈ π‘ˆ)
 
Theoremcatstr 17911 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
{⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩} Struct ⟨1, 15⟩
 
Theoremfucval 17912* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (𝐢 Func 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &   π΄ = (Baseβ€˜πΆ)    &    Β· = (compβ€˜π·)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    &   (πœ‘ β†’ βˆ™ = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))    β‡’   (πœ‘ β†’ 𝑄 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), π‘βŸ©, ⟨(compβ€˜ndx), βˆ™ ⟩})
 
Theoremfuccofval 17913* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (𝐢 Func 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &   π΄ = (Baseβ€˜πΆ)    &    Β· = (compβ€˜π·)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    &    βˆ™ = (compβ€˜π‘„)    β‡’   (πœ‘ β†’ βˆ™ = (𝑣 ∈ (𝐡 Γ— 𝐡), β„Ž ∈ 𝐡 ↦ ⦋(1st β€˜π‘£) / π‘“β¦Œβ¦‹(2nd β€˜π‘£) / π‘”β¦Œ(𝑏 ∈ (π‘”π‘β„Ž), π‘Ž ∈ (𝑓𝑁𝑔) ↦ (π‘₯ ∈ 𝐴 ↦ ((π‘β€˜π‘₯)(⟨((1st β€˜π‘“)β€˜π‘₯), ((1st β€˜π‘”)β€˜π‘₯)⟩ Β· ((1st β€˜β„Ž)β€˜π‘₯))(π‘Žβ€˜π‘₯))))))
 
Theoremfucbas 17914 The objects of the functor category are functors from 𝐢 to 𝐷. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    β‡’   (𝐢 Func 𝐷) = (Baseβ€˜π‘„)
 
Theoremfuchom 17915 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    β‡’   π‘ = (Hom β€˜π‘„)
 
TheoremfuchomOLD 17916 Obsolete proof of fuchom 17915 as of 14-Oct-2024. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    β‡’   π‘ = (Hom β€˜π‘„)
 
Theoremfucco 17917* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &   π΄ = (Baseβ€˜πΆ)    &    Β· = (compβ€˜π·)    &    βˆ™ = (compβ€˜π‘„)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    &   (πœ‘ β†’ 𝑆 ∈ (𝐺𝑁𝐻))    β‡’   (πœ‘ β†’ (𝑆(⟨𝐹, 𝐺⟩ βˆ™ 𝐻)𝑅) = (π‘₯ ∈ 𝐴 ↦ ((π‘†β€˜π‘₯)(⟨((1st β€˜πΉ)β€˜π‘₯), ((1st β€˜πΊ)β€˜π‘₯)⟩ Β· ((1st β€˜π»)β€˜π‘₯))(π‘…β€˜π‘₯))))
 
Theoremfuccoval 17918 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &   π΄ = (Baseβ€˜πΆ)    &    Β· = (compβ€˜π·)    &    βˆ™ = (compβ€˜π‘„)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    &   (πœ‘ β†’ 𝑆 ∈ (𝐺𝑁𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ ((𝑆(⟨𝐹, 𝐺⟩ βˆ™ 𝐻)𝑅)β€˜π‘‹) = ((π‘†β€˜π‘‹)(⟨((1st β€˜πΉ)β€˜π‘‹), ((1st β€˜πΊ)β€˜π‘‹)⟩ Β· ((1st β€˜π»)β€˜π‘‹))(π‘…β€˜π‘‹)))
 
Theoremfuccocl 17919 The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &    βˆ™ = (compβ€˜π‘„)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    &   (πœ‘ β†’ 𝑆 ∈ (𝐺𝑁𝐻))    β‡’   (πœ‘ β†’ (𝑆(⟨𝐹, 𝐺⟩ βˆ™ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
 
Theoremfucidcl 17920 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &    1 = (Idβ€˜π·)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    β‡’   (πœ‘ β†’ ( 1 ∘ (1st β€˜πΉ)) ∈ (𝐹𝑁𝐹))
 
Theoremfuclid 17921 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &    βˆ™ = (compβ€˜π‘„)    &    1 = (Idβ€˜π·)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    β‡’   (πœ‘ β†’ (( 1 ∘ (1st β€˜πΊ))(⟨𝐹, 𝐺⟩ βˆ™ 𝐺)𝑅) = 𝑅)
 
Theoremfucrid 17922 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &    βˆ™ = (compβ€˜π‘„)    &    1 = (Idβ€˜π·)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    β‡’   (πœ‘ β†’ (𝑅(⟨𝐹, 𝐹⟩ βˆ™ 𝐺)( 1 ∘ (1st β€˜πΉ))) = 𝑅)
 
Theoremfucass 17923 Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π‘ = (𝐢 Nat 𝐷)    &    βˆ™ = (compβ€˜π‘„)    &   (πœ‘ β†’ 𝑅 ∈ (𝐹𝑁𝐺))    &   (πœ‘ β†’ 𝑆 ∈ (𝐺𝑁𝐻))    &   (πœ‘ β†’ 𝑇 ∈ (𝐻𝑁𝐾))    β‡’   (πœ‘ β†’ ((𝑇(⟨𝐺, 𝐻⟩ βˆ™ 𝐾)𝑆)(⟨𝐹, 𝐺⟩ βˆ™ 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻⟩ βˆ™ 𝐾)(𝑆(⟨𝐹, 𝐺⟩ βˆ™ 𝐻)𝑅)))
 
Theoremfuccatid 17924* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    &    1 = (Idβ€˜π·)    β‡’   (πœ‘ β†’ (𝑄 ∈ Cat ∧ (Idβ€˜π‘„) = (𝑓 ∈ (𝐢 Func 𝐷) ↦ ( 1 ∘ (1st β€˜π‘“)))))
 
Theoremfuccat 17925 The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    β‡’   (πœ‘ β†’ 𝑄 ∈ Cat)
 
Theoremfucid 17926 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   πΌ = (Idβ€˜π‘„)    &    1 = (Idβ€˜π·)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    β‡’   (πœ‘ β†’ (πΌβ€˜πΉ) = ( 1 ∘ (1st β€˜πΉ)))
 
Theoremfucsect 17927* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (Baseβ€˜πΆ)    &   π‘ = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    &   (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))    &   π‘† = (Sectβ€˜π‘„)    &   π‘‡ = (Sectβ€˜π·)    β‡’   (πœ‘ β†’ (π‘ˆ(𝐹𝑆𝐺)𝑉 ↔ (π‘ˆ ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝑇((1st β€˜πΊ)β€˜π‘₯))(π‘‰β€˜π‘₯))))
 
Theoremfucinv 17928* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (Baseβ€˜πΆ)    &   π‘ = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    &   (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))    &   πΌ = (Invβ€˜π‘„)    &   π½ = (Invβ€˜π·)    β‡’   (πœ‘ β†’ (π‘ˆ(𝐹𝐼𝐺)𝑉 ↔ (π‘ˆ ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ βˆ€π‘₯ ∈ 𝐡 (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))(π‘‰β€˜π‘₯))))
 
Theoreminvfuc 17929* If 𝑉(π‘₯) is an inverse to π‘ˆ(π‘₯) for each π‘₯, and π‘ˆ is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (Baseβ€˜πΆ)    &   π‘ = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    &   (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))    &   πΌ = (Invβ€˜π‘„)    &   π½ = (Invβ€˜π·)    &   (πœ‘ β†’ π‘ˆ ∈ (𝐹𝑁𝐺))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘ˆβ€˜π‘₯)(((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯))𝑋)    β‡’   (πœ‘ β†’ π‘ˆ(𝐹𝐼𝐺)(π‘₯ ∈ 𝐡 ↦ 𝑋))
 
Theoremfuciso 17930* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑄 = (𝐢 FuncCat 𝐷)    &   π΅ = (Baseβ€˜πΆ)    &   π‘ = (𝐢 Nat 𝐷)    &   (πœ‘ β†’ 𝐹 ∈ (𝐢 Func 𝐷))    &   (πœ‘ β†’ 𝐺 ∈ (𝐢 Func 𝐷))    &   πΌ = (Isoβ€˜π‘„)    &   π½ = (Isoβ€˜π·)    β‡’   (πœ‘ β†’ (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ βˆ€π‘₯ ∈ 𝐡 (π΄β€˜π‘₯) ∈ (((1st β€˜πΉ)β€˜π‘₯)𝐽((1st β€˜πΊ)β€˜π‘₯)))))
 
Theoremnatpropd 17931 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
(πœ‘ β†’ (Homf β€˜π΄) = (Homf β€˜π΅))    &   (πœ‘ β†’ (compfβ€˜π΄) = (compfβ€˜π΅))    &   (πœ‘ β†’ (Homf β€˜πΆ) = (Homf β€˜π·))    &   (πœ‘ β†’ (compfβ€˜πΆ) = (compfβ€˜π·))    &   (πœ‘ β†’ 𝐴 ∈ Cat)    &   (πœ‘ β†’ 𝐡 ∈ Cat)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    β‡’   (πœ‘ β†’ (𝐴 Nat 𝐢) = (𝐡 Nat 𝐷))
 
Theoremfucpropd 17932 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
(πœ‘ β†’ (Homf β€˜π΄) = (Homf β€˜π΅))    &   (πœ‘ β†’ (compfβ€˜π΄) = (compfβ€˜π΅))    &   (πœ‘ β†’ (Homf β€˜πΆ) = (Homf β€˜π·))    &   (πœ‘ β†’ (compfβ€˜πΆ) = (compfβ€˜π·))    &   (πœ‘ β†’ 𝐴 ∈ Cat)    &   (πœ‘ β†’ 𝐡 ∈ Cat)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐷 ∈ Cat)    β‡’   (πœ‘ β†’ (𝐴 FuncCat 𝐢) = (𝐡 FuncCat 𝐷))
 
8.1.10  Initial, terminal and zero objects of a category
 
Syntaxcinito 17933 Extend class notation with the class of initial objects of a category.
class InitO
 
Syntaxctermo 17934 Extend class notation with the class of terminal objects of a category.
class TermO
 
Syntaxczeroo 17935 Extend class notation with the class of zero objects of a category.
class ZeroO
 
Definitiondf-inito 17936* An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in [Adamek] p. 101, or definition in [Lang] p. 57 (called "a universally repelling object" there). See dfinito2 17955 and dfinito3 17957 for alternate definitions depending on df-termo 17937. (Contributed by AV, 3-Apr-2020.)
InitO = (𝑐 ∈ Cat ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏)})
 
Definitiondf-termo 17937* An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). See dftermo2 17956 and dftermo3 17958 for alternate definitions depending on df-inito 17936. (Contributed by AV, 3-Apr-2020.)
TermO = (𝑐 ∈ Cat ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž)})
 
Definitiondf-zeroo 17938 An object A is called a zero object provided that it is both an initial object and a terminal object. Definition 7.7 of [Adamek] p. 103. (Contributed by AV, 3-Apr-2020.)
ZeroO = (𝑐 ∈ Cat ↦ ((InitOβ€˜π‘) ∩ (TermOβ€˜π‘)))
 
Theoreminitofn 17939 InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO Fn Cat
 
Theoremtermofn 17940 TermO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO Fn Cat
 
Theoremzeroofn 17941 ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
ZeroO Fn Cat
 
Theoreminitorcl 17942 Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝐼 ∈ (InitOβ€˜πΆ) β†’ 𝐢 ∈ Cat)
 
Theoremtermorcl 17943 Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝑇 ∈ (TermOβ€˜πΆ) β†’ 𝐢 ∈ Cat)
 
Theoremzeroorcl 17944 Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020.)
(𝑍 ∈ (ZeroOβ€˜πΆ) β†’ 𝐢 ∈ Cat)
 
Theoreminitoval 17945* The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   π΅ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    β‡’   (πœ‘ β†’ (InitOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)})
 
Theoremtermoval 17946* The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   π΅ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    β‡’   (πœ‘ β†’ (TermOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)})
 
Theoremzerooval 17947 The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   π΅ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    β‡’   (πœ‘ β†’ (ZeroOβ€˜πΆ) = ((InitOβ€˜πΆ) ∩ (TermOβ€˜πΆ)))
 
Theoremisinito 17948* The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐼 ∈ (InitOβ€˜πΆ) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
 
Theoremistermo 17949* The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐼 ∈ (TermOβ€˜πΆ) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑏𝐻𝐼)))
 
Theoremiszeroo 17950 The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐼 ∈ (ZeroOβ€˜πΆ) ↔ (𝐼 ∈ (InitOβ€˜πΆ) ∧ 𝐼 ∈ (TermOβ€˜πΆ))))
 
Theoremisinitoi 17951* Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    β‡’   ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) β†’ (𝑂 ∈ 𝐡 ∧ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑏)))
 
Theoremistermoi 17952* Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    β‡’   ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ (𝑂 ∈ 𝐡 ∧ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑏𝐻𝑂)))
 
Theoreminitoid 17953 For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    β‡’   ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)})
 
Theoremtermoid 17954 For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
𝐡 = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    β‡’   ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)})
 
Theoremdfinito2 17955 An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17936 depending on df-termo 17937. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (𝑐 ∈ Cat ↦ (TermOβ€˜(oppCatβ€˜π‘)))
 
Theoremdftermo2 17956 A terminal object is an initial object in the opposite category. An alternate definition of df-termo 17937 depending on df-inito 17936. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (𝑐 ∈ Cat ↦ (InitOβ€˜(oppCatβ€˜π‘)))
 
Theoremdfinito3 17957 An alternate definition of df-inito 17936 depending on df-termo 17937, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (TermO ∘ (oppCat β†Ύ Cat))
 
Theoremdftermo3 17958 An alternate definition of df-termo 17937 depending on df-inito 17936, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (InitO ∘ (oppCat β†Ύ Cat))
 
Theoreminitoo 17959 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
(𝐢 ∈ Cat β†’ (𝑂 ∈ (InitOβ€˜πΆ) β†’ 𝑂 ∈ (Baseβ€˜πΆ)))
 
Theoremtermoo 17960 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
(𝐢 ∈ Cat β†’ (𝑂 ∈ (TermOβ€˜πΆ) β†’ 𝑂 ∈ (Baseβ€˜πΆ)))
 
Theoremiszeroi 17961 Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.)
((𝐢 ∈ Cat ∧ 𝑂 ∈ (ZeroOβ€˜πΆ)) β†’ (𝑂 ∈ (Baseβ€˜πΆ) ∧ (𝑂 ∈ (InitOβ€˜πΆ) ∧ 𝑂 ∈ (TermOβ€˜πΆ))))
 
Theorem2initoinv 17962 Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))    β‡’   ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺)
 
Theoreminitoeu1 17963* Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))    β‡’   (πœ‘ β†’ βˆƒ!𝑓 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡))
 
Theoreminitoeu1w 17964 Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))    β‡’   (πœ‘ β†’ 𝐴( ≃𝑐 β€˜πΆ)𝐡)
 
Theoreminitoeu2lem0 17965 Lemma 0 for initoeu2 17968. (Contributed by AV, 9-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   π‘‹ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   πΌ = (Isoβ€˜πΆ)    &    ⚬ = (compβ€˜πΆ)    β‡’   (((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) ∧ ((𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ)) = (𝐺(⟨𝐴, 𝐡⟩ ⚬ 𝐷)((𝐡(Invβ€˜πΆ)𝐴)β€˜πΎ))) β†’ 𝐺 = (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾))
 
Theoreminitoeu2lem1 17966* Lemma 1 for initoeu2 17968. (Contributed by AV, 9-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   π‘‹ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   πΌ = (Isoβ€˜πΆ)    &    ⚬ = (compβ€˜πΆ)    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐡𝐻𝐷))) β†’ ((βˆƒ!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐡𝐻𝐷)) β†’ 𝐺 = (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾)))
 
Theoreminitoeu2lem2 17967* Lemma 2 for initoeu2 17968. (Contributed by AV, 10-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   π‘‹ = (Baseβ€˜πΆ)    &   π» = (Hom β€˜πΆ)    &   πΌ = (Isoβ€˜πΆ)    &    ⚬ = (compβ€˜πΆ)    β‡’   ((πœ‘ ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐡𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐡, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐡𝐻𝐷))) β†’ (βˆƒ!𝑓 𝑓 ∈ (𝐴𝐻𝐷) β†’ βˆƒ!𝑔 𝑔 ∈ (𝐡𝐻𝐷)))
 
Theoreminitoeu2 17968 Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (InitOβ€˜πΆ))    &   (πœ‘ β†’ 𝐴( ≃𝑐 β€˜πΆ)𝐡)    β‡’   (πœ‘ β†’ 𝐡 ∈ (InitOβ€˜πΆ))
 
Theorem2termoinv 17969 Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (TermOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (TermOβ€˜πΆ))    β‡’   ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺)
 
Theoremtermoeu1 17970* Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 18-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (TermOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (TermOβ€˜πΆ))    β‡’   (πœ‘ β†’ βˆƒ!𝑓 𝑓 ∈ (𝐴(Isoβ€˜πΆ)𝐡))
 
Theoremtermoeu1w 17971 Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ Cat)    &   (πœ‘ β†’ 𝐴 ∈ (TermOβ€˜πΆ))    &   (πœ‘ β†’ 𝐡 ∈ (TermOβ€˜πΆ))    β‡’   (πœ‘ β†’ 𝐴( ≃𝑐 β€˜πΆ)𝐡)
 
8.2  Arrows (disjointified hom-sets)
 
Syntaxcdoma 17972 Extend class notation to include the domain extractor for an arrow.
class doma
 
Syntaxccoda 17973 Extend class notation to include the codomain extractor for an arrow.
class coda
 
Syntaxcarw 17974 Extend class notation to include the collection of all arrows of a category.
class Arrow
 
Syntaxchoma 17975 Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa
 
Definitiondf-doma 17976 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
doma = (1st ∘ 1st )
 
Definitiondf-coda 17977 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda = (2nd ∘ 1st )
 
Definitiondf-homa 17978* Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 17976 and df-coda 17977. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa = (𝑐 ∈ Cat ↦ (π‘₯ ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) ↦ ({π‘₯} Γ— ((Hom β€˜π‘)β€˜π‘₯))))
 
Definitiondf-arw 17979 Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to Hom, which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
Arrow = (𝑐 ∈ Cat ↦ βˆͺ ran (Homaβ€˜π‘))
 
Theoremhomarcl 17980 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ 𝐢 ∈ Cat)
 
Theoremhomafval 17981* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   π½ = (Hom β€˜πΆ)    β‡’   (πœ‘ β†’ 𝐻 = (π‘₯ ∈ (𝐡 Γ— 𝐡) ↦ ({π‘₯} Γ— (π½β€˜π‘₯))))
 
Theoremhomaf 17982 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    β‡’   (πœ‘ β†’ 𝐻:(𝐡 Γ— 𝐡)βŸΆπ’« ((𝐡 Γ— 𝐡) Γ— V))
 
Theoremhomaval 17983 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   π½ = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘‹π»π‘Œ) = ({βŸ¨π‘‹, π‘ŒβŸ©} Γ— (π‘‹π½π‘Œ)))
 
Theoremelhoma 17984 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   π½ = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑍(π‘‹π»π‘Œ)𝐹 ↔ (𝑍 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝐹 ∈ (π‘‹π½π‘Œ))))
 
Theoremelhomai 17985 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   π½ = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ (π‘‹π½π‘Œ))    β‡’   (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ©(π‘‹π»π‘Œ)𝐹)
 
Theoremelhomai2 17986 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝐢 ∈ Cat)    &   π½ = (Hom β€˜πΆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ (π‘‹π½π‘Œ))    β‡’   (πœ‘ β†’ βŸ¨π‘‹, π‘Œ, 𝐹⟩ ∈ (π‘‹π»π‘Œ))
 
Theoremhomarcl2 17987 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
 
Theoremhomarel 17988 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   Rel (π‘‹π»π‘Œ)
 
Theoremhoma1 17989 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   (𝑍(π‘‹π»π‘Œ)𝐹 β†’ 𝑍 = βŸ¨π‘‹, π‘ŒβŸ©)
 
Theoremhomahom2 17990 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π½ = (Hom β€˜πΆ)    β‡’   (𝑍(π‘‹π»π‘Œ)𝐹 β†’ 𝐹 ∈ (π‘‹π½π‘Œ))
 
Theoremhomahom 17991 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    &   π½ = (Hom β€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (2nd β€˜πΉ) ∈ (π‘‹π½π‘Œ))
 
Theoremhomadm 17992 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (domaβ€˜πΉ) = 𝑋)
 
Theoremhomacd 17993 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (codaβ€˜πΉ) = π‘Œ)
 
Theoremhomadmcd 17994 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homaβ€˜πΆ)    β‡’   (𝐹 ∈ (π‘‹π»π‘Œ) β†’ 𝐹 = βŸ¨π‘‹, π‘Œ, (2nd β€˜πΉ)⟩)
 
Theoremarwval 17995 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    &   π» = (Homaβ€˜πΆ)    β‡’   π΄ = βˆͺ ran 𝐻
 
Theoremarwrcl 17996 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝐢 ∈ Cat)
 
Theoremarwhoma 17997 An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    &   π» = (Homaβ€˜πΆ)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝐹 ∈ ((domaβ€˜πΉ)𝐻(codaβ€˜πΉ)))
 
Theoremhomarw 17998 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    &   π» = (Homaβ€˜πΆ)    β‡’   (π‘‹π»π‘Œ) βŠ† 𝐴
 
Theoremarwdm 17999 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    β‡’   (𝐹 ∈ 𝐴 β†’ (domaβ€˜πΉ) ∈ 𝐡)
 
Theoremarwcd 18000 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrowβ€˜πΆ)    &   π΅ = (Baseβ€˜πΆ)    β‡’   (𝐹 ∈ 𝐴 β†’ (codaβ€˜πΉ) ∈ 𝐡)
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