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Theorem itg2l 19084
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 2347 . 2  |-  ( A  e.  L  <->  A  e.  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g ) ) } )
3 simpr 447 . . . . 5  |-  ( ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  =  ( S.1 `  g ) )
4 fvex 5539 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2371 . . . 4  |-  ( ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2663 . . 3  |-  ( E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2289 . . . . 5  |-  ( x  =  A  ->  (
x  =  ( S.1 `  g )  <->  A  =  ( S.1 `  g ) ) )
87anbi2d 684 . . . 4  |-  ( x  =  A  ->  (
( g  o R  <_  F  /\  x  =  ( S.1 `  g
) )  <->  ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
98rexbidv 2564 . . 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 2921 . 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 240 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   dom cdm 4689   ` cfv 5255    o Rcofr 6077    <_ cle 8868   S.1citg1 18970
This theorem is referenced by:  itg2lr  19085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263
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