Theorem idahom

Description: Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	idafval.i	⊢ 𝐼 = ( Ida ‘ 𝐶 )
	idafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	idafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	idahom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	idahom.h	⊢ 𝐻 = ( Homa ‘ 𝐶 )
Assertion	idahom	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		idafval.i	⊢ 𝐼 = ( Ida ‘ 𝐶 )
2		idafval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		idafval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		idahom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		idahom.h	⊢ 𝐻 = ( Homa ‘ 𝐶 )
6		eqid	⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
7	1 2 3 6 4	idaval	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) = ⟨ 𝑋 , 𝑋 , ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ⟩ )
8		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
9	2 8 6 3 4	catidcl	⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
10	5 2 3 8 4 4 9	elhomai2	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑋 , ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ⟩ ∈ ( 𝑋 𝐻 𝑋 ) )
11	7 10	eqeltrd	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) )