Metamath Proof Explorer


Theorem i1fmbf

Description: Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014)

Ref Expression
Assertion i1fmbf ( 𝐹 ∈ dom ∫1𝐹 ∈ MblFn )

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 simplbi ( 𝐹 ∈ dom ∫1𝐹 ∈ MblFn )