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Theorem indval 27517
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 27516 . . 3  |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
21adantr 465 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
(𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
3 eleq2 2533 . . . . 5  |-  ( a  =  A  ->  (
x  e.  a  <->  x  e.  A ) )
43ifbid 3954 . . . 4  |-  ( a  =  A  ->  if ( x  e.  a ,  1 ,  0 )  =  if ( x  e.  A , 
1 ,  0 ) )
54mpteq2dv 4527 . . 3  |-  ( a  =  A  ->  (
x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
65adantl 466 . 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 461 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  C_  O )
8 ssexg 4586 . . . . 5  |-  ( ( A  C_  O  /\  O  e.  V )  ->  A  e.  _V )
98ancoms 453 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  e.  _V )
10 elpwg 4011 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ~P O  <->  A 
C_  O ) )
119, 10syl 16 . . 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 6121 . . 3  |-  ( O  e.  V  ->  (
x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  _V )
1413adantr 465 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) )  e. 
_V )
152, 6, 12, 14fvmptd 5946 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 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ifcif 3932   ~Pcpw 4003    |-> cmpt 4498   ` cfv 5579   0cc0 9481   1c1 9482  𝟭cind 27514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ind 27515
This theorem is referenced by:  indval2  27518  indf  27519  indfval  27520
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