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Metamath Proof Explorer

Theorem catidcl

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

Ref Expression
Hypotheses catidcl.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
catidcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
catidcl.i ⊢ 1 = ( Id ‘ 𝐶 )
catidcl.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
catidcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion catidcl ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )

Proof

Step Hyp Ref Expression
1 catidcl.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 catidcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
3 catidcl.i ⊢ 1 = ( Id ‘ 𝐶 )
4 catidcl.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
5 catidcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 eqid ⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
7 1 2 6 4 3 5 cidval ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
8 1 2 6 4 5 catideu ⊢ ( 𝜑 → ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) )
9 riotacl ⊢ ( ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ ( 𝑋 𝐻 𝑋 ) )
10 8 9 syl ⊢ ( 𝜑 → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( ⟨ 𝑦 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ ( 𝑋 𝐻 𝑋 ) )
11 7 10 eqeltrd ⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )