Theorem idfu1st

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 )
Assertion	idfu1st	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		idfuval.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2		idfuval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		idfuval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
5	1 2 3 4	idfuval	⊢ ( 𝜑 → 𝐼 = ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝐶 ) ‘ 𝑧 ) ) ) ⟩ )
6	5	fveq2d	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( 1st ‘ ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝐶 ) ‘ 𝑧 ) ) ) ⟩ ) )
7	2	fvexi	⊢ 𝐵 ∈ V
8		resiexg	⊢ ( 𝐵 ∈ V → ( I ↾ 𝐵 ) ∈ V )
9	7 8	ax-mp	⊢ ( I ↾ 𝐵 ) ∈ V
10	7 7	xpex	⊢ ( 𝐵 × 𝐵 ) ∈ V
11	10	mptex	⊢ ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝐶 ) ‘ 𝑧 ) ) ) ∈ V
12	9 11	op1st	⊢ ( 1st ‘ ⟨ ( I ↾ 𝐵 ) , ( 𝑧 ∈ ( 𝐵 × 𝐵 ) ↦ ( I ↾ ( ( Hom ‘ 𝐶 ) ‘ 𝑧 ) ) ) ⟩ ) = ( I ↾ 𝐵 )
13	6 12	eqtrdi	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )