Theorem idfu1

Description: Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	idfuval.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
	idfuval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	idfuval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	idfu1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	idfu1	⊢ ( 𝜑 → ( ( 1st ‘ 𝐼 ) ‘ 𝑋 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		idfuval.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2		idfuval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		idfuval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		idfu1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5	1 2 3	idfu1st	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )
6	5	fveq1d	⊢ ( 𝜑 → ( ( 1st ‘ 𝐼 ) ‘ 𝑋 ) = ( ( I ↾ 𝐵 ) ‘ 𝑋 ) )
7		fvresi	⊢ ( 𝑋 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑋 ) = 𝑋 )
8	4 7	syl	⊢ ( 𝜑 → ( ( I ↾ 𝐵 ) ‘ 𝑋 ) = 𝑋 )
9	6 8	eqtrd	⊢ ( 𝜑 → ( ( 1st ‘ 𝐼 ) ‘ 𝑋 ) = 𝑋 )