Theorem natfn2

Description: A natural transformation is a function on the objects of the source category. (Contributed by Zhi Wang, 14-Sep-2026)

	Ref	Expression
Hypotheses	isnat3.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
	isnat3.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	natfn2.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) )
Assertion	natfn2	⊢ ( 𝜑 → 𝐴 Fn 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		isnat3.n	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
2		isnat3.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		natfn2.a	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) )
4	1 3	nat1st2nd	⊢ ( 𝜑 → 𝐴 ∈ ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ 𝑁 ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ ) )
5	1 4 2	natfn	⊢ ( 𝜑 → 𝐴 Fn 𝐵 )