Theorem afvvv

Description: If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvvv	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		afvprc	⊢ ( ¬ 𝐴 ∈ V → ( 𝐹 ''' 𝐴 ) = V )
2		nvelim	⊢ ( ( 𝐹 ''' 𝐴 ) = V → ¬ ( 𝐹 ''' 𝐴 ) ∈ 𝐵 )
3	1 2	syl	⊢ ( ¬ 𝐴 ∈ V → ¬ ( 𝐹 ''' 𝐴 ) ∈ 𝐵 )
4	3	con4i	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → 𝐴 ∈ V )