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Theorem itg2l 23541
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 2722 . 2 (𝐴𝐿𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))})
3 simpr 476 . . . . 5 ((𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 = (∫1𝑔))
4 fvex 6239 . . . . 5 (∫1𝑔) ∈ V
53, 4syl6eqel 2738 . . . 4 ((𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
65rexlimivw 3058 . . 3 (∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
7 eqeq1 2655 . . . . 5 (𝑥 = 𝐴 → (𝑥 = (∫1𝑔) ↔ 𝐴 = (∫1𝑔)))
87anbi2d 740 . . . 4 (𝑥 = 𝐴 → ((𝑔𝑟𝐹𝑥 = (∫1𝑔)) ↔ (𝑔𝑟𝐹𝐴 = (∫1𝑔))))
98rexbidv 3081 . . 3 (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔))))
106, 9elab3 3390 . 2 (𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))} ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)))
112, 10bitri 264 1 (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝐴 = (∫1𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231   class class class wbr 4685  dom cdm 5143  cfv 5926  𝑟 cofr 6938  cle 10113  1citg1 23429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-fv 5934
This theorem is referenced by:  itg2lr  23542
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