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Theorem indpi1 27663
Description: Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 21826. (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 27662 . . . . 5  |-  ( ( O  e.  V  /\  A  C_  O  /\  x  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  =  1  <-> 
x  e.  A ) )
213expia 1193 . . . 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 27657 . . . 4  |-  ( ( O  e.  V  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
5 ffn 5724 . . . 4  |-  ( ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  A
)  Fn  O )
6 fniniseg 5995 . . . 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 3493 . . . . 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 2459 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 1374    e. wcel 1762    C_ wss 3471   {csn 4022   {cpr 4024   `'ccnv 4993   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581   0cc0 9483   1c1 9484  𝟭cind 27652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-i2m1 9551  ax-1ne0 9552  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-ind 27653
This theorem is referenced by:  indf1ofs  27667  eulerpartlemgf  27946
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