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Theorem indval 26608
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 26607 . . 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 2524 . . . . 5  |-  ( a  =  A  ->  (
x  e.  a  <->  x  e.  A ) )
43ifbid 3912 . . . 4  |-  ( a  =  A  ->  if ( x  e.  a ,  1 ,  0 )  =  if ( x  e.  A , 
1 ,  0 ) )
54mpteq2dv 4480 . . 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 4539 . . . . 5  |-  ( ( A  C_  O  /\  O  e.  V )  ->  A  e.  _V )
98ancoms 453 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  ->  A  e.  _V )
10 elpwg 3969 . . . 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 6049 . . 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 5881 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 1370    e. wcel 1758   _Vcvv 3071    C_ wss 3429   ifcif 3892   ~Pcpw 3961    |-> cmpt 4451   ` cfv 5519   0cc0 9386   1c1 9387  𝟭cind 26605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ind 26606
This theorem is referenced by:  indval2  26609  indf  26610  indfval  26611
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