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Theorem ditgex 19700
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 19478 . 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 19623 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9268 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3765 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2482 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2924   ifcif 3707   class class class wbr 4180  (class class class)co 6048    <_ cle 9085   -ucneg 9256   (,)cioo 10880   S.citg 19471   S__cdit 19472
This theorem is referenced by:  itgsubstlem  19893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-uni 3984  df-iota 5385  df-fv 5429  df-ov 6051  df-neg 9258  df-sum 12443  df-itg 19477  df-ditg 19478
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