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Theorem indv 28823
Description: Value of the indicator function generator with domain  O. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Assertion
Ref Expression
indv  |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
Distinct variable groups:    x, a, O    V, a
Allowed substitution hint:    V( x)

Proof of Theorem indv
Dummy variable  o is distinct from all other variables.
StepHypRef Expression
1 df-ind 28822 . . 3  |- 𝟭  =  ( o  e.  _V  |->  ( a  e.  ~P o  |->  ( x  e.  o 
|->  if ( x  e.  a ,  1 ,  0 ) ) ) )
21a1i 11 . 2  |-  ( O  e.  V  -> 𝟭  =  ( o  e.  _V  |->  ( a  e.  ~P o  |->  ( x  e.  o 
|->  if ( x  e.  a ,  1 ,  0 ) ) ) ) )
3 pweq 3979 . . . 4  |-  ( o  =  O  ->  ~P o  =  ~P O
)
4 mpteq1 4498 . . . 4  |-  ( o  =  O  ->  (
x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) )  =  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )
53, 4mpteq12dv 4496 . . 3  |-  ( o  =  O  ->  (
a  e.  ~P o  |->  ( x  e.  o 
|->  if ( x  e.  a ,  1 ,  0 ) ) )  =  ( a  e. 
~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
65adantl 467 . 2  |-  ( ( O  e.  V  /\  o  =  O )  ->  ( a  e.  ~P o  |->  ( x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) ) )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
7 elex 3087 . 2  |-  ( O  e.  V  ->  O  e.  _V )
8 pwexg 4601 . . 3  |-  ( O  e.  _V  ->  ~P O  e.  _V )
9 mptexg 6142 . . 3  |-  ( ~P O  e.  _V  ->  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )  e.  _V )
107, 8, 93syl 18 . 2  |-  ( O  e.  V  ->  (
a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )  e.  _V )
112, 6, 7, 10fvmptd 5962 1  |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078   ifcif 3906   ~Pcpw 3976    |-> cmpt 4476   ` cfv 5593   0cc0 9535   1c1 9536  𝟭cind 28821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ind 28822
This theorem is referenced by:  indval  28824  indf1o  28834
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