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 )