Theorem afv2fv0b

Description: The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2fv0b	⊢ ( ( 𝐹 ‘ 𝐴 ) = ∅ ↔ ( ( 𝐹 '''' 𝐴 ) = ∅ ∨ ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		afv2fv0	⊢ ( ( 𝐹 ‘ 𝐴 ) = ∅ → ( ( 𝐹 '''' 𝐴 ) = ∅ ∨ ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ) )
2		afv20fv0	⊢ ( ( 𝐹 '''' 𝐴 ) = ∅ → ( 𝐹 ‘ 𝐴 ) = ∅ )
3		afv2ndeffv0	⊢ ( ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 → ( 𝐹 ‘ 𝐴 ) = ∅ )
4	2 3	jaoi	⊢ ( ( ( 𝐹 '''' 𝐴 ) = ∅ ∨ ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ) → ( 𝐹 ‘ 𝐴 ) = ∅ )
5	1 4	impbii	⊢ ( ( 𝐹 ‘ 𝐴 ) = ∅ ↔ ( ( 𝐹 '''' 𝐴 ) = ∅ ∨ ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ) )