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Theorem itg2l 22428
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 2480 . 2  |-  ( A  e.  L  <->  A  e.  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) } )
3 simpr 459 . . . . 5  |-  ( ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  =  ( S.1 `  g ) )
4 fvex 5859 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2498 . . . 4  |-  ( ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2893 . . 3  |-  ( E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2406 . . . . 5  |-  ( x  =  A  ->  (
x  =  ( S.1 `  g )  <->  A  =  ( S.1 `  g ) ) )
87anbi2d 702 . . . 4  |-  ( x  =  A  ->  (
( g  oR  <_  F  /\  x  =  ( S.1 `  g
) )  <->  ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) ) ) )
98rexbidv 2918 . . 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 3203 . 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 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2755   _Vcvv 3059   class class class wbr 4395   dom cdm 4823   ` cfv 5569    oRcofr 6520    <_ cle 9659   S.1citg1 22316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-sn 3973  df-pr 3975  df-uni 4192  df-iota 5533  df-fv 5577
This theorem is referenced by:  itg2lr  22429
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