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Theorem indpi1 28254
Description: Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 22265. (Contributed by Thierry Arnoux, 21-Aug-2017.)
Assertion
Ref Expression
indpi1  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( `' ( (𝟭 `  O ) `  A
) " { 1 } )  =  A )

Proof of Theorem indpi1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ind1a 28253 . . . . 5  |-  ( ( O  e.  V  /\  A  C_  O  /\  x  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  =  1  <-> 
x  e.  A ) )
213expia 1196 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  O  ->  ( ( ( (𝟭 `  O ) `  A
) `  x )  =  1  <->  x  e.  A ) ) )
32pm5.32d 637 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( ( x  e.  O  /\  ( ( (𝟭 `  O ) `  A ) `  x
)  =  1 )  <-> 
( x  e.  O  /\  x  e.  A
) ) )
4 indf 28248 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
5 ffn 5713 . . . 4  |-  ( ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  A
)  Fn  O )
6 fniniseg 5984 . . . 4  |-  ( ( (𝟭 `  O ) `  A )  Fn  O  ->  ( x  e.  ( `' ( (𝟭 `  O
) `  A ) " { 1 } )  <-> 
( x  e.  O  /\  ( ( (𝟭 `  O
) `  A ) `  x )  =  1 ) ) )
74, 5, 63syl 20 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  ( `' ( (𝟭 `  O
) `  A ) " { 1 } )  <-> 
( x  e.  O  /\  ( ( (𝟭 `  O
) `  A ) `  x )  =  1 ) ) )
8 ssel 3483 . . . . 5  |-  ( A 
C_  O  ->  (
x  e.  A  ->  x  e.  O )
)
98pm4.71rd 633 . . . 4  |-  ( A 
C_  O  ->  (
x  e.  A  <->  ( x  e.  O  /\  x  e.  A ) ) )
109adantl 464 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  A  <->  ( x  e.  O  /\  x  e.  A )
) )
113, 7, 103bitr4d 285 . 2  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  ( `' ( (𝟭 `  O
) `  A ) " { 1 } )  <-> 
x  e.  A ) )
1211eqrdv 2451 1  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( `' ( (𝟭 `  O ) `  A
) " { 1 } )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   {csn 4016   {cpr 4018   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570   0cc0 9481   1c1 9482  𝟭cind 28243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-ind 28244
This theorem is referenced by:  indf1ofs  28258  eulerpartlemgf  28585
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