Theorem idadm

Description: Domain 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 )
Assertion	idadm	⊢ ( 𝜑 → ( doma ‘ ( 𝐼 ‘ 𝑋 ) ) = 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		eqid	⊢ ( Homa ‘ 𝐶 ) = ( Homa ‘ 𝐶 )
6	1 2 3 4 5	idahom	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝑋 ( Homa ‘ 𝐶 ) 𝑋 ) )
7	5	homadm	⊢ ( ( 𝐼 ‘ 𝑋 ) ∈ ( 𝑋 ( Homa ‘ 𝐶 ) 𝑋 ) → ( doma ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 )
8	6 7	syl	⊢ ( 𝜑 → ( doma ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 )