Theorem fexd

Description: If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	fexd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
	fexd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
Assertion	fexd	⊢ ( 𝜑 → 𝐹 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		fexd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fexd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
3		fex	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ 𝐶 ) → 𝐹 ∈ V )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐹 ∈ V )