Theorem fndmexb

Description: The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024)

	Ref	Expression
Assertion	fndmexb	⊢ ( 𝐹 Fn 𝐴 → ( 𝐴 ∈ V ↔ 𝐹 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		fnex	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ V ) → 𝐹 ∈ V )
2	1	ex	⊢ ( 𝐹 Fn 𝐴 → ( 𝐴 ∈ V → 𝐹 ∈ V ) )
3		simpr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ) → 𝐹 ∈ V )
4		simpl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ) → 𝐹 Fn 𝐴 )
5	3 4	fndmexd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 ∈ V ) → 𝐴 ∈ V )
6	5	ex	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ∈ V → 𝐴 ∈ V ) )
7	2 6	impbid	⊢ ( 𝐹 Fn 𝐴 → ( 𝐴 ∈ V ↔ 𝐹 ∈ V ) )