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Theorem itg2l 23302
Description: Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))}
Assertion
Ref Expression
itg2l (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)))
Distinct variable groups:   𝑥,𝑔,𝐴   𝑔,𝐹,𝑥
Allowed substitution hints:   𝐿(𝑥,𝑔)

Proof of Theorem itg2l
StepHypRef Expression
1 itg2val.1 . . 3 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))}
21eleq2i 2680 . 2 (𝐴𝐿𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))})
3 simpr 476 . . . . 5 ((𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 = (∫1𝑔))
4 fvex 6113 . . . . 5 (∫1𝑔) ∈ V
53, 4syl6eqel 2696 . . . 4 ((𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
65rexlimivw 3011 . . 3 (∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
7 eqeq1 2614 . . . . 5 (𝑥 = 𝐴 → (𝑥 = (∫1𝑔) ↔ 𝐴 = (∫1𝑔)))
87anbi2d 736 . . . 4 (𝑥 = 𝐴 → ((𝑔𝑟𝐹𝑥 = (∫1𝑔)) ↔ (𝑔𝑟𝐹𝐴 = (∫1𝑔))))
98rexbidv 3034 . . 3 (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔))))
106, 9elab3 3327 . 2 (𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)))
112, 10bitri 263 1 (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  Vcvv 3173   class class class wbr 4583  dom cdm 5038  cfv 5804  𝑟 cofr 6794  cle 9954  1citg1 23190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-fv 5812
This theorem is referenced by:  itg2lr  23303
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