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Theorem ditgex 21445
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 21440 . 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 21366 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9711 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 3958 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2535 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   _Vcvv 3070   ifcif 3891   class class class wbr 4392  (class class class)co 6192    <_ cle 9522   -ucneg 9699   (,)cioo 11403   S.citg 21216   S__cdit 21439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4521
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-uni 4192  df-iota 5481  df-fv 5526  df-ov 6195  df-neg 9701  df-sum 13268  df-itg 21221  df-ditg 21440
This theorem is referenced by:  itgsubstlem  21638
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