Theorem ovn0dmfun

Description: If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 . (Contributed by AV, 27-Jan-2020)

	Ref	Expression
Assertion	ovn0dmfun	|- ( ( A F B ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ov	|- ( A F B ) = ( F ` <. A , B >. )
2	1	neeq1i	|- ( ( A F B ) =/= (/) <-> ( F ` <. A , B >. ) =/= (/) )
3		fvfundmfvn0	|- ( ( F ` <. A , B >. ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )
4	2 3	sylbi	|- ( ( A F B ) =/= (/) -> ( <. A , B >. e. dom F /\ Fun ( F |` { <. A , B >. } ) ) )