Theorem afvnufveq

Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvnufveq	⊢ ( ( 𝐹 ''' 𝐴 ) ≠ V → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		afvfundmfveq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
2		afvnfundmuv	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 ''' 𝐴 ) = V )
3	1 2	nsyl5	⊢ ( ¬ ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) → ( 𝐹 ''' 𝐴 ) = V )
4	3	necon1ai	⊢ ( ( 𝐹 ''' 𝐴 ) ≠ V → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )