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Theorem indval 28461
Description: Value of the indicator function generator for a set  A and a domain  O. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Assertion
Ref Expression
indval  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
Distinct variable groups:    x, O    x, A
Allowed substitution hint:    V( x)

Proof of Theorem indval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 indv 28460 . . 3  |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
21adantr 463 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
(𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
3 eleq2 2475 . . . . 5  |-  ( a  =  A  ->  (
x  e.  a  <->  x  e.  A ) )
43ifbid 3907 . . . 4  |-  ( a  =  A  ->  if ( x  e.  a ,  1 ,  0 )  =  if ( x  e.  A , 
1 ,  0 ) )
54mpteq2dv 4482 . . 3  |-  ( a  =  A  ->  (
x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
65adantl 464 . 2  |-  ( ( ( O  e.  V  /\  A  C_  O )  /\  a  =  A )  ->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) )  =  ( x  e.  O  |->  if ( x  e.  A , 
1 ,  0 ) ) )
7 simpr 459 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  C_  O )
8 ssexg 4540 . . . . 5  |-  ( ( A  C_  O  /\  O  e.  V )  ->  A  e.  _V )
98ancoms 451 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  e.  _V )
10 elpwg 3963 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ~P O  <->  A 
C_  O ) )
119, 10syl 17 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( A  e.  ~P O 
<->  A  C_  O )
)
127, 11mpbird 232 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  e.  ~P O
)
13 mptexg 6123 . . 3  |-  ( O  e.  V  ->  (
x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  _V )
1413adantr 463 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) )  e. 
_V )
152, 6, 12, 14fvmptd 5938 1  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   ifcif 3885   ~Pcpw 3955    |-> cmpt 4453   ` cfv 5569   0cc0 9522   1c1 9523  𝟭cind 28458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ind 28459
This theorem is referenced by:  indval2  28462  indf  28463  indfval  28464
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