Theorem idfu1stalem

Description: Lemma for idfu1sta . (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	idfu1stalem	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )

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	1 2	eqeltrrid	⊢ ( 𝜑 → ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) )
5		idfurcl	⊢ ( ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
6	1	idfucl	⊢ ( 𝐶 ∈ Cat → 𝐼 ∈ ( 𝐶 Func 𝐶 ) )
7	4 5 6	3syl	⊢ ( 𝜑 → 𝐼 ∈ ( 𝐶 Func 𝐶 ) )
8	7	func1st2nd	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) ( 𝐶 Func 𝐶 ) ( 2nd ‘ 𝐼 ) )
9	2	func1st2nd	⊢ ( 𝜑 → ( 1st ‘ 𝐼 ) ( 𝐷 Func 𝐸 ) ( 2nd ‘ 𝐼 ) )
10	8 9	funchomf	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
11	10	homfeqbas	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
12	3 11	eqtr4d	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )