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Theorem itg2l 19611
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 2499 . 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 5734 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2523 . . . 4  |-  ( ( g  o R  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2818 . . 3  |-  ( E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2441 . . . . 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 2718 . . 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 3081 . 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948   class class class wbr 4204   dom cdm 4870   ` cfv 5446    o Rcofr 6296    <_ cle 9111   S.1citg1 19497
This theorem is referenced by:  itg2lr  19612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410  df-fv 5454
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