Theorem afv0fv0

Description: If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afv0fv0	⊢ ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ‘ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		0ex	⊢ ∅ ∈ V
2		eleq1a	⊢ ( ∅ ∈ V → ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ''' 𝐴 ) ∈ V ) )
3	1 2	ax-mp	⊢ ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ''' 𝐴 ) ∈ V )
4		afvvfveq	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ V → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
5		eqeq1	⊢ ( ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) → ( ( 𝐹 ''' 𝐴 ) = ∅ ↔ ( 𝐹 ‘ 𝐴 ) = ∅ ) )
6	5	biimpd	⊢ ( ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) → ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ‘ 𝐴 ) = ∅ ) )
7	4 6	syl	⊢ ( ( 𝐹 ''' 𝐴 ) ∈ V → ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ‘ 𝐴 ) = ∅ ) )
8	3 7	mpcom	⊢ ( ( 𝐹 ''' 𝐴 ) = ∅ → ( 𝐹 ‘ 𝐴 ) = ∅ )