Theorem fnmptf

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013) (Revised by Thierry Arnoux, 10-May-2017)

	Ref	Expression
Hypothesis	mptfnf.0	⊢ Ⅎ 𝑥 𝐴
Assertion	fnmptf	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mptfnf.0	⊢ Ⅎ 𝑥 𝐴
2		elex	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ V )
3	2	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V )
4	1	mptfnf	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 )
5	3 4	sylib	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 )