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Theorem itg2l 19490
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  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2l  |-  ( A  e.  L  <->  E. g  e.  dom  S.1 ( g  o R  <_  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  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }
21eleq2i 2453 . 2  |-  ( A  e.  L  <->  A  e.  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) } )
3 simpr 448 . . . . 5  |-  ( ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  =  ( S.1 `  g ) )
4 fvex 5684 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2477 . . . 4  |-  ( ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2771 . . 3  |-  ( E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2395 . . . . 5  |-  ( x  =  A  ->  (
x  =  ( S.1 `  g )  <->  A  =  ( S.1 `  g ) ) )
87anbi2d 685 . . . 4  |-  ( x  =  A  ->  (
( g  o R  <_  F  /\  x  =  ( S.1 `  g
) )  <->  ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
98rexbidv 2672 . . 3  |-  ( x  =  A  ->  ( E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
106, 9elab3 3034 . 2  |-  ( A  e.  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
) ) }  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) ) )
112, 10bitri 241 1  |-  ( A  e.  L  <->  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2375   E.wrex 2652   _Vcvv 2901   class class class wbr 4155   dom cdm 4820   ` cfv 5396    o Rcofr 6245    <_ cle 9056   S.1citg1 19376
This theorem is referenced by:  itg2lr  19491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-nul 4281
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-sn 3765  df-pr 3766  df-uni 3960  df-iota 5360  df-fv 5404
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