Description: A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | i1frn | ⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin ) |
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 ) |