Theorem fvn0fvelrn

Description: If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018) (Proof shortened by SN, 13-Jan-2025)

	Ref	Expression
Assertion	fvn0fvelrn	⊢ ( ( 𝐹 ‘ 𝑋 ) ≠ ∅ → ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		fvrn0	⊢ ( 𝐹 ‘ 𝑋 ) ∈ ( ran 𝐹 ∪ { ∅ } )
2		nelsn	⊢ ( ( 𝐹 ‘ 𝑋 ) ≠ ∅ → ¬ ( 𝐹 ‘ 𝑋 ) ∈ { ∅ } )
3		elunnel2	⊢ ( ( ( 𝐹 ‘ 𝑋 ) ∈ ( ran 𝐹 ∪ { ∅ } ) ∧ ¬ ( 𝐹 ‘ 𝑋 ) ∈ { ∅ } ) → ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 )
4	1 2 3	sylancr	⊢ ( ( 𝐹 ‘ 𝑋 ) ≠ ∅ → ( 𝐹 ‘ 𝑋 ) ∈ ran 𝐹 )