Theorem 0func

Description: The functor from the empty category. (Contributed by Zhi Wang, 7-Oct-2025) (Proof shortened by Zhi Wang, 17-Oct-2025)

	Ref	Expression
Hypothesis	0func.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
Assertion	0func	⊢ ( 𝜑 → ( ∅ Func 𝐶 ) = { ⟨ ∅ , ∅ ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		0func.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
2		0ex	⊢ ∅ ∈ V
3	2	a1i	⊢ ( 𝜑 → ∅ ∈ V )
4		base0	⊢ ∅ = ( Base ‘ ∅ )
5	4	a1i	⊢ ( 𝜑 → ∅ = ( Base ‘ ∅ ) )
6	3 5 1	0funcg	⊢ ( 𝜑 → ( ∅ Func 𝐶 ) = { ⟨ ∅ , ∅ ⟩ } )