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Theorem itg2l 24442
 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 2843 . 2 (𝐴𝐿𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))})
3 simpr 488 . . . . 5 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 = (∫1𝑔))
4 fvex 6676 . . . . 5 (∫1𝑔) ∈ V
53, 4eqeltrdi 2860 . . . 4 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
65rexlimivw 3206 . . 3 (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
7 eqeq1 2762 . . . . 5 (𝑥 = 𝐴 → (𝑥 = (∫1𝑔) ↔ 𝐴 = (∫1𝑔)))
87anbi2d 631 . . . 4 (𝑥 = 𝐴 → ((𝑔r𝐹𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝐴 = (∫1𝑔))))
98rexbidv 3221 . . 3 (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔))))
106, 9elab3 3597 . 2 (𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))
112, 10bitri 278 1 (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071  Vcvv 3409   class class class wbr 5036  dom cdm 5528  ‘cfv 6340   ∘r cofr 7410   ≤ cle 10727  ∫1citg1 24328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-uni 4802  df-iota 6299  df-fv 6348 This theorem is referenced by:  itg2lr  24443
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