Theorem aovvoveq

Description: The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovvoveq	|- ( (( A F B )) e. C -> (( A F B )) = ( A F 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		afvvfveq	|- ( ( F ''' <. A , B >. ) e. C -> ( F ''' <. A , B >. ) = ( F ` <. A , B >. ) )
4		df-ov	|- ( A F B ) = ( F ` <. A , B >. )
5	3 1 4	3eqtr4g	|- ( ( F ''' <. A , B >. ) e. C -> (( A F B )) = ( A F B ) )
6	2 5	sylbi	|- ( (( A F B )) e. C -> (( A F B )) = ( A F B ) )