Theorem fnmptd

Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	fnmptd.1	⊢ Ⅎ 𝑥 𝜑
	fnmptd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	fnmptd.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	fnmptd	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		fnmptd.1	⊢ Ⅎ 𝑥 𝜑
2		fnmptd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
3		fnmptd.3	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
4	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉 ) )
5	1 4	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 )
6	3	fnmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴 )
7	5 6	syl	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )