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Theorem ditgex 19770
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
Assertion
Ref Expression
ditgex  |-  S__ [ A  ->  B ] C  _d x  e.  _V

Proof of Theorem ditgex
StepHypRef Expression
1 df-ditg 19765 . 2  |-  S__ [ A  ->  B ] C  _d x  =  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )
2 itgex 19691 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9335 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3821 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2512 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1727   _Vcvv 2962   ifcif 3763   class class class wbr 4237  (class class class)co 6110    <_ cle 9152   -ucneg 9323   (,)cioo 10947   S.citg 19541   S__cdit 19764
This theorem is referenced by:  itgsubstlem  19963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-nul 4363
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-uni 4040  df-iota 5447  df-fv 5491  df-ov 6113  df-neg 9325  df-sum 12511  df-itg 19546  df-ditg 19765
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