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Theorem itg2l 21309
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  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2l  |-  ( A  e.  L  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) )
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  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
21eleq2i 2526 . 2  |-  ( A  e.  L  <->  A  e.  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) } )
3 simpr 461 . . . . 5  |-  ( ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  =  ( S.1 `  g ) )
4 fvex 5785 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2544 . . . 4  |-  ( ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2919 . . 3  |-  ( E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2453 . . . . 5  |-  ( x  =  A  ->  (
x  =  ( S.1 `  g )  <->  A  =  ( S.1 `  g ) ) )
87anbi2d 703 . . . 4  |-  ( x  =  A  ->  (
( g  oR  <_  F  /\  x  =  ( S.1 `  g
) )  <->  ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
98rexbidv 2816 . . 3  |-  ( x  =  A  ->  ( E. g  e.  dom  S.1 ( g  oR  <_  F  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
106, 9elab3 3196 . 2  |-  ( A  e.  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  x  =  ( S.1 `  g
) ) }  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) )
112, 10bitri 249 1  |-  ( A  e.  L  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   {cab 2435   E.wrex 2793   _Vcvv 3054   class class class wbr 4376   dom cdm 4924   ` cfv 5502    oRcofr 6405    <_ cle 9506   S.1citg1 21197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-nul 4505
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-sn 3962  df-pr 3964  df-uni 4176  df-iota 5465  df-fv 5510
This theorem is referenced by:  itg2lr  21310
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