Theorem idfu2

Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017)

	Ref	Expression
Hypotheses	idfuval.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
	idfuval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	idfuval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	idfuval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	idfu2nd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	idfu2nd.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	idfu2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
Assertion	idfu2	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		idfuval.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2		idfuval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		idfuval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		idfuval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		idfu2nd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		idfu2nd.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		idfu2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
8	1 2 3 4 5 6	idfu2nd	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ ( 𝑋 𝐻 𝑌 ) ) )
9	8	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) )
10		fvresi	⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) = 𝐹 )
11	7 10	syl	⊢ ( 𝜑 → ( ( I ↾ ( 𝑋 𝐻 𝑌 ) ) ‘ 𝐹 ) = 𝐹 )
12	9 11	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) ‘ 𝐹 ) = 𝐹 )