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Theorem ditgex 19417
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 19195 . 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 19340 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9197 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3712 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2436 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1715   _Vcvv 2873   ifcif 3654   class class class wbr 4125  (class class class)co 5981    <_ cle 9015   -ucneg 9185   (,)cioo 10809   S.citg 19188   S__cdit 19189
This theorem is referenced by:  itgsubstlem  19610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-nul 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-uni 3930  df-iota 5322  df-fv 5366  df-ov 5984  df-neg 9187  df-sum 12367  df-itg 19194  df-ditg 19195
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