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Theorem indv 26322
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 26321 . . 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 3851 . . . 4  |-  ( o  =  O  ->  ~P o  =  ~P O
)
4 mpteq1 4360 . . . 4  |-  ( o  =  O  ->  (
x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) )  =  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )
53, 4mpteq12dv 4358 . . 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 463 . 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 2971 . 2  |-  ( O  e.  V  ->  O  e.  _V )
8 pwexg 4464 . . 3  |-  ( O  e.  _V  ->  ~P O  e.  _V )
9 mptexg 5934 . . 3  |-  ( ~P O  e.  _V  ->  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )  e.  _V )
107, 8, 93syl 20 . 2  |-  ( O  e.  V  ->  (
a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) )  e.  _V )
112, 6, 7, 10fvmptd 5767 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 1362    e. wcel 1755   _Vcvv 2962   ifcif 3779   ~Pcpw 3848    e. cmpt 4338   ` cfv 5406   0cc0 9269   1c1 9270  𝟭cind 26320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ind 26321
This theorem is referenced by:  indval  26323  indf1o  26333
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