Theorem afvvfunressn

Description: If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvvfunressn	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → Fun ( 𝐹 ↾ { 𝐴 } ) )

Proof

Step	Hyp	Ref	Expression
1		nfunsnafv	⊢ ( ¬ Fun ( 𝐹 ↾ { 𝐴 } ) → ( 𝐹 ''' 𝐴 ) = V )
2		nvelim	⊢ ( ( 𝐹 ''' 𝐴 ) = V → ¬ ( 𝐹 ''' 𝐴 ) ∈ 𝐵 )
3	1 2	syl	⊢ ( ¬ Fun ( 𝐹 ↾ { 𝐴 } ) → ¬ ( 𝐹 ''' 𝐴 ) ∈ 𝐵 )
4	3	con4i	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → Fun ( 𝐹 ↾ { 𝐴 } ) )