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Theorem indpi1 27908
Description: Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 21969. (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 27907 . . . . 5  |-  ( ( O  e.  V  /\  A  C_  O  /\  x  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  =  1  <-> 
x  e.  A ) )
213expia 1199 . . . 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 27902 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
5 ffn 5721 . . . 4  |-  ( ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  A
)  Fn  O )
6 fniniseg 5993 . . . 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 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 1383    e. wcel 1804    C_ wss 3461   {csn 4014   {cpr 4016   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578   0cc0 9495   1c1 9496  𝟭cind 27897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-i2m1 9563  ax-1ne0 9564  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-ind 27898
This theorem is referenced by:  indf1ofs  27912  eulerpartlemgf  28191
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