Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ind1a Structured version   Unicode version

Theorem ind1a 28678
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 28674 . . 3  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( (𝟭 `  O ) `  A ) `  X
)  =  if ( X  e.  A , 
1 ,  0 ) )
21eqeq1d 2422 . 2  |-  ( ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  X )  =  1  <-> 
if ( X  e.  A ,  1 ,  0 )  =  1 ) )
3 eqid 2420 . . . . 5  |-  1  =  1
43biantru 507 . . . 4  |-  ( X  e.  A  <->  ( X  e.  A  /\  1  =  1 ) )
5 ax-1ne0 9597 . . . . . 6  |-  1  =/=  0
65neii 2620 . . . . 5  |-  -.  1  =  0
76biorfi 408 . . . 4  |-  ( ( X  e.  A  /\  1  =  1 )  <-> 
( ( X  e.  A  /\  1  =  1 )  \/  1  =  0 ) )
86bianfi 933 . . . . 5  |-  ( 1  =  0  <->  ( -.  X  e.  A  /\  1  =  0 ) )
98orbi2i 521 . . . 4  |-  ( ( ( X  e.  A  /\  1  =  1
)  \/  1  =  0 )  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
104, 7, 93bitri 274 . . 3  |-  ( X  e.  A  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
11 eqif 3944 . . 3  |-  ( 1  =  if ( X  e.  A ,  1 ,  0 )  <->  ( ( X  e.  A  /\  1  =  1 )  \/  ( -.  X  e.  A  /\  1  =  0 ) ) )
12 eqcom 2429 . . 3  |-  ( 1  =  if ( X  e.  A ,  1 ,  0 )  <->  if ( X  e.  A , 
1 ,  0 )  =  1 )
1310, 11, 123bitr2ri 277 . 2  |-  ( if ( X  e.  A ,  1 ,  0 )  =  1  <->  X  e.  A )
142, 13syl6bb 264 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433   ifcif 3906   ` cfv 5592   0cc0 9528   1c1 9529  𝟭cind 28668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-i2m1 9596  ax-1ne0 9597  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-ind 28669
This theorem is referenced by:  indpi1  28679  indpreima  28682
  Copyright terms: Public domain W3C validator