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Theorem ind1a 28282
Description: Value of the indicator function where it is  1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
Assertion
Ref Expression
ind1a  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  X )  =  1  <-> 
X  e.  A ) )

Proof of Theorem ind1a
StepHypRef Expression
1 indfval 28278 . . 3  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( (𝟭 `  O ) `  A ) `  X
)  =  if ( X  e.  A , 
1 ,  0 ) )
21eqeq1d 2459 . 2  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  X )  =  1  <-> 
if ( X  e.  A ,  1 ,  0 )  =  1 ) )
3 eqid 2457 . . . . 5  |-  1  =  1
43biantru 505 . . . 4  |-  ( X  e.  A  <->  ( X  e.  A  /\  1  =  1 ) )
5 ax-1ne0 9578 . . . . . 6  |-  1  =/=  0
65neii 2656 . . . . 5  |-  -.  1  =  0
76biorfi 407 . . . 4  |-  ( ( X  e.  A  /\  1  =  1 )  <-> 
( ( X  e.  A  /\  1  =  1 )  \/  1  =  0 ) )
86bianfi 925 . . . . 5  |-  ( 1  =  0  <->  ( -.  X  e.  A  /\  1  =  0 ) )
98orbi2i 519 . . . 4  |-  ( ( ( X  e.  A  /\  1  =  1
)  \/  1  =  0 )  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
104, 7, 93bitri 271 . . 3  |-  ( X  e.  A  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
11 eqif 3982 . . 3  |-  ( 1  =  if ( X  e.  A ,  1 ,  0 )  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
12 eqcom 2466 . . 3  |-  ( 1  =  if ( X  e.  A ,  1 ,  0 )  <->  if ( X  e.  A , 
1 ,  0 )  =  1 )
1310, 11, 123bitr2ri 274 . 2  |-  ( if ( X  e.  A ,  1 ,  0 )  =  1  <->  X  e.  A )
142, 13syl6bb 261 1  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  X )  =  1  <-> 
X  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ifcif 3944   ` cfv 5594   0cc0 9509   1c1 9510  𝟭cind 28272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-ind 28273
This theorem is referenced by:  indpi1  28283  indpreima  28286
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