Theorem afvfundmfveq

Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvfundmfveq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dfafv2	⊢ ( 𝐹 ''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , V )
2		iftrue	⊢ ( 𝐹 defAt 𝐴 → if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , V ) = ( 𝐹 ‘ 𝐴 ) )
3	1 2	syl5eq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )