Theorem fex

Description: If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999)

	Ref	Expression
Assertion	fex	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → 𝐹 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2		fnex	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐹 ∈ V )
3	1 2	sylan	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → 𝐹 ∈ V )