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 ) )