Theorem fnmpt

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013)

	Ref	Expression
Hypothesis	mptfng.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	fnmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mptfng.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		elex	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ V )
3	2	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V )
4	1	mptfng	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴 )
5	3 4	sylib	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴 )