MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2l Structured version   Visualization version   GIF version

Theorem itg2l 25117
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 2826 . 2 (𝐴𝐿𝐴 ∈ {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))})
3 simpr 486 . . . . 5 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 = (∫1𝑔))
4 fvex 6859 . . . . 5 (∫1𝑔) ∈ V
53, 4eqeltrdi 2842 . . . 4 ((𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
65rexlimivw 3145 . . 3 (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)) → 𝐴 ∈ V)
7 eqeq1 2737 . . . . 5 (𝑥 = 𝐴 → (𝑥 = (∫1𝑔) ↔ 𝐴 = (∫1𝑔)))
87anbi2d 630 . . . 4 (𝑥 = 𝐴 → ((𝑔r𝐹𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝐴 = (∫1𝑔))))
98rexbidv 3172 . . 3 (𝑥 = 𝐴 → (∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔))))
106, 9elab3 3642 . 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 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3070  Vcvv 3447   class class class wbr 5109  dom cdm 5637  cfv 6500  r cofr 7620  cle 11198  1citg1 25002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rex 3071  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-pr 4593  df-uni 4870  df-iota 6452  df-fv 6508
This theorem is referenced by:  itg2lr  25118
  Copyright terms: Public domain W3C validator