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Theorem itg2l 25583
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 2817 . 2 (𝐴𝐿𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))})
3 simpr 484 . . . . 5 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 = (∫1𝑔))
4 fvex 6895 . . . . 5 (∫1𝑔) ∈ V
53, 4eqeltrdi 2833 . . . 4 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
65rexlimivw 3143 . . 3 (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
7 eqeq1 2728 . . . . 5 (𝑥 = 𝐴 → (𝑥 = (∫1𝑔) ↔ 𝐴 = (∫1𝑔)))
87anbi2d 628 . . . 4 (𝑥 = 𝐴 → ((𝑔r𝐹𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝐴 = (∫1𝑔))))
98rexbidv 3170 . . 3 (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔))))
106, 9elab3 3669 . 2 (𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))
112, 10bitri 275 1 (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  {cab 2701  wrex 3062  Vcvv 3466   class class class wbr 5139  dom cdm 5667  cfv 6534  r cofr 7663  cle 11247  1citg1 25468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rex 3063  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-sn 4622  df-pr 4624  df-uni 4901  df-iota 6486  df-fv 6542
This theorem is referenced by:  itg2lr  25584
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