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Theorem indpi1 26477
Description: Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 21166. (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 26476 . . . . 5  |-  ( ( O  e.  V  /\  A  C_  O  /\  x  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  =  1  <-> 
x  e.  A ) )
213expia 1189 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( x  e.  O  ->  ( ( ( (𝟭 `  O ) `  A
) `  x )  =  1  <->  x  e.  A ) ) )
32pm5.32d 639 . . 3  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( ( x  e.  O  /\  ( ( (𝟭 `  O ) `  A ) `  x
)  =  1 )  <-> 
( x  e.  O  /\  x  e.  A
) ) )
4 indf 26471 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
5 ffn 5558 . . . 4  |-  ( ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  A
)  Fn  O )
6 fniniseg 5823 . . . 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 3349 . . . . 5  |-  ( A 
C_  O  ->  (
x  e.  A  ->  x  e.  O )
)
98pm4.71rd 635 . . . 4  |-  ( A 
C_  O  ->  (
x  e.  A  <->  ( x  e.  O  /\  x  e.  A ) ) )
109adantl 466 . . 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 2440 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 369    = wceq 1369    e. wcel 1756    C_ wss 3327   {csn 3876   {cpr 3878   `'ccnv 4838   "cima 4842    Fn wfn 5412   -->wf 5413   ` cfv 5417   0cc0 9281   1c1 9282  𝟭cind 26466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-i2m1 9349  ax-1ne0 9350  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-ind 26467
This theorem is referenced by:  indf1ofs  26481  eulerpartlemgf  26761
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