Theorem ditgex 25894

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

Assertion

Ref	Expression
ditgex	⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V

Proof of Theorem ditgex

Step	Hyp	Ref	Expression
1		df-ditg 25889	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2		itgex 25812	. . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V
3		negex 11425	. . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V
4	2, 3	ifex 4530	. 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V
5	1, 4	eqeltri 2857	1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V

Syntax hints:   ∈ wcel 2141  Vcvv 3453  ifcif 4479   class class class wbr 5099  (class class class)co 7392   ≤ cle 11214  -cneg 11412  (,)cioo 13346  ∫citg 25660  ⨜cdit 25888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473  df-fv 6525  df-ov 7395  df-neg 11414  df-sum 15697  df-itg 25665  df-ditg 25889

This theorem is referenced by:  itgsubstlem  26090