Theorem aovvfunressn

Description: If the operation 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, 26-May-2017)

	Ref	Expression
Assertion	aovvfunressn	|- ( (( A F B )) e. C -> Fun ( F |` { <. A , B >. } ) )

Proof

Step	Hyp	Ref	Expression
1		df-aov	|- (( A F B )) = ( F ''' <. A , B >. )
2	1	eleq1i	|- ( (( A F B )) e. C <-> ( F ''' <. A , B >. ) e. C )
3		afvvfunressn	|- ( ( F ''' <. A , B >. ) e. C -> Fun ( F |` { <. A , B >. } ) )
4	2 3	sylbi	|- ( (( A F B )) e. C -> Fun ( F |` { <. A , B >. } ) )