Theorem idfu1sta

Description: Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025)

	Ref	Expression
Hypotheses	idfu2nda.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
	idfu2nda.d	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐷 Func 𝐸 ) )
	idfu2nda.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐷 ) )
Assertion	idfu1sta	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		idfu2nda.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2		idfu2nda.d	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐷 Func 𝐸 ) )
3		idfu2nda.b	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐷 ) )
4		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
5	1 2	eqeltrrid	⊢ ( 𝜑 → ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) )
6		idfurcl	⊢ ( ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
7	5 6	syl	⊢ ( 𝜑 → 𝐶 ∈ Cat )
8	1 4 7	idfu1st	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ ( Base ‘ 𝐶 ) ) )
9	1 2 3	idfu1stalem	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
10	9	reseq2d	⊢ ( 𝜑 → ( I ↾ 𝐵 ) = ( I ↾ ( Base ‘ 𝐶 ) ) )
11	8 10	eqtr4d	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )