Metamath Proof Explorer


Theorem i1frn

Description: A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014)

Ref Expression
Assertion i1frn ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin )

Proof

Step Hyp Ref Expression
1 isi1f ( 𝐹 ∈ dom ∫1 ↔ ( 𝐹 ∈ MblFn ∧ ( 𝐹 : ℝ ⟶ ℝ ∧ ran 𝐹 ∈ Fin ∧ ( vol ‘ ( 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) ) )
2 1 simprbi ( 𝐹 ∈ dom ∫1 → ( 𝐹 : ℝ ⟶ ℝ ∧ ran 𝐹 ∈ Fin ∧ ( vol ‘ ( 𝐹 “ ( ℝ ∖ { 0 } ) ) ) ∈ ℝ ) )
3 2 simp2d ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin )