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Metamath Proof Explorer

Theorem fnexd

Description: If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses fnexd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
fnexd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion fnexd ( 𝜑 → 𝐹 ∈ V )

Proof

Step Hyp Ref Expression
1 fnexd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2 fnexd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3 fnex ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐹 ∈ V )
4 1 2 3 syl2anc ⊢ ( 𝜑 → 𝐹 ∈ V )