Theorem ditgex

Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014)

	Ref	Expression
Assertion	ditgex	⊢ ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-ditg	⊢ ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = if ( 𝐴 ≤ 𝐵 , ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 , - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 )
2		itgex	⊢ ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 ∈ V
3		negex	⊢ - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ∈ V
4	2 3	ifex	⊢ if ( 𝐴 ≤ 𝐵 , ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 , - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) ∈ V
5	1 4	eqeltri	⊢ ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 ∈ V