Theorem idfu2nda

Description: Value of the morphism 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 )
	idfu2nda.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	idfu2nda.h	⊢ ( 𝜑 → 𝐻 = ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) )
Assertion	idfu2nda	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ 𝐻 ) )

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		idfu2nda.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		idfu2nda.h	⊢ ( 𝜑 → 𝐻 = ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) )
7		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
8	1 2	eqeltrrid	⊢ ( 𝜑 → ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) )
9		idfurcl	⊢ ( ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
10	8 9	syl	⊢ ( 𝜑 → 𝐶 ∈ Cat )
11		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
12	1 2 3	idfu1stalem	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
13	4 12	eleqtrd	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
14	5 12	eleqtrd	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
15	1 7 10 11 13 14	idfu2nd	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
16		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
17	1	idfucl	⊢ ( 𝐶 ∈ Cat → 𝐼 ∈ ( 𝐶 Func 𝐶 ) )
18	10 17	syl	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐶 Func 𝐶 ) )
19	18	func1st2nd	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) ( 𝐶 Func 𝐶 ) ( 2nd ‘ 𝐼 ) )
20	2	func1st2nd	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) ( 𝐷 Func 𝐸 ) ( 2nd ‘ 𝐼 ) )
21	19 20	funchomf	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
22	7 11 16 21 13 14	homfeqval	⊢ ( 𝜑 → ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) )
23	6 22	eqtr4d	⊢ ( 𝜑 → 𝐻 = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
24	23	reseq2d	⊢ ( 𝜑 → ( I ↾ 𝐻 ) = ( I ↾ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
25	15 24	eqtr4d	⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐼 ) 𝑌 ) = ( I ↾ 𝐻 ) )