Theorem itgex 25671

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

Assertion

Ref	Expression
itgex	⊢ ∫𝐴𝐵 d𝑥 ∈ V

Proof of Theorem itgex

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-itg 25524	. 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2		sumex 15654	. 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) ∈ V
3	1, 2	eqeltri 2824	1 ⊢ ∫𝐴𝐵 d𝑥 ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2109  Vcvv 3447  ⦋csb 3862  ifcif 4488   class class class wbr 5107   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  ℝcr 11067  0cc0 11068  ici 11070   · cmul 11073   ≤ cle 11209   / cdiv 11835  3c3 12242  ...cfz 13468  ↑cexp 14026  ℜcre 15063  Σcsu 15652  ∫₂citg2 25517  ∫citg 25519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-sn 4590  df-pr 4592  df-uni 4872  df-iota 6464  df-sum 15653  df-itg 25524

This theorem is referenced by:  ditgex  25753  ftc1lem1  25942  itgulm  26317  dmarea  26867  dfarea  26870  areaval  26874  ftc1anc  37695  itgsinexp  45953  wallispilem1  46063  wallispilem2  46064