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Theorem ditgex 21991
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 21986 . 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 21912 . . 3  |-  S. ( A (,) B ) C  _d x  e. 
_V
3 negex 9814 . . 3  |-  -u S. ( B (,) A ) C  _d x  e. 
_V
42, 3ifex 4008 . 2  |-  if ( A  <_  B ,  S. ( A (,) B
) C  _d x ,  -u S. ( B (,) A ) C  _d x )  e. 
_V
51, 4eqeltri 2551 1  |-  S__ [ A  ->  B ] C  _d x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   _Vcvv 3113   ifcif 3939   class class class wbr 4447  (class class class)co 6282    <_ cle 9625   -ucneg 9802   (,)cioo 11525   S.citg 21762   S__cdit 21985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-uni 4246  df-iota 5549  df-fv 5594  df-ov 6285  df-neg 9804  df-sum 13468  df-itg 21767  df-ditg 21986
This theorem is referenced by:  itgsubstlem  22184
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