Theorem itgex

Description: An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014)

	Ref	Expression
Assertion	itgex	⊢ ∫ 𝐴 𝐵 d 𝑥 ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-itg	⊢ ∫ 𝐴 𝐵 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) )
2		sumex	⊢ Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) ∈ V
3	1 2	eqeltri	⊢ ∫ 𝐴 𝐵 d 𝑥 ∈ V