Theorem afvvfveq

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	afvvfveq	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		nvelim	⊢ ( ( 𝐹 ''' 𝐴 ) = V → ¬ ( 𝐹 ''' 𝐴 ) ∈ 𝐵 )
2	1	necon2ai	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → ( 𝐹 ''' 𝐴 ) ≠ V )
3		afvnufveq	⊢ ( ( 𝐹 ''' 𝐴 ) ≠ V → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
4	2 3	syl	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ 𝐵 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )