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Theorem itg2l 21868
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 2545 . 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 5874 . . . . 5  |-  ( S.1 `  g )  e.  _V
53, 4syl6eqel 2563 . . . 4  |-  ( ( g  oR  <_  F  /\  A  =  ( S.1 `  g ) )  ->  A  e.  _V )
65rexlimivw 2952 . . 3  |-  ( E. g  e.  dom  S.1 ( g  oR  <_  F  /\  A  =  ( S.1 `  g
) )  ->  A  e.  _V )
7 eqeq1 2471 . . . . 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 2973 . . 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 3257 . 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 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   dom cdm 4999   ` cfv 5586    oRcofr 6521    <_ cle 9625   S.1citg1 21756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5549  df-fv 5594
This theorem is referenced by:  itg2lr  21869
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