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Theorem ditgex 22129
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 22124 . 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 22050 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9823 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3995 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2527 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1804   _Vcvv 3095   ifcif 3926   class class class wbr 4437  (class class class)co 6281    <_ cle 9632   -ucneg 9811   (,)cioo 11538   S.citg 21900   S__cdit 22123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-uni 4235  df-iota 5541  df-fv 5586  df-ov 6284  df-neg 9813  df-sum 13488  df-itg 21905  df-ditg 22124
This theorem is referenced by:  itgsubstlem  22322
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