Theorem idfu1a

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 ) )
	idfu2nda.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	idfu1a	⊢ ( 𝜑 → ( ( 1st ‘ 𝐼 ) ‘ 𝑋 ) = 𝑋 )

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		idfu2nda.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
6	1 2	eqeltrrid	⊢ ( 𝜑 → ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) )
7		idfurcl	⊢ ( ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
8	6 7	syl	⊢ ( 𝜑 → 𝐶 ∈ Cat )
9	1 2 3	idfu1stalem	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
10	4 9	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
11	1 5 8 10	idfu1	⊢ ( 𝜑 → ( ( 1st ‘ 𝐼 ) ‘ 𝑋 ) = 𝑋 )