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Theorem indpreima 26493
Description: A function with range  { 0 ,  1 } as an indicator of the preimage of  { 1 } (Contributed by Thierry Arnoux, 23-Aug-2017.)
Assertion
Ref Expression
indpreima  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  =  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) )

Proof of Theorem indpreima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ffn 5571 . . 3  |-  ( F : O --> { 0 ,  1 }  ->  F  Fn  O )
21adantl 466 . 2  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  Fn  O
)
3 cnvimass 5201 . . . . 5  |-  ( `' F " { 1 } )  C_  dom  F
4 fdm 5575 . . . . . 6  |-  ( F : O --> { 0 ,  1 }  ->  dom 
F  =  O )
54adantl 466 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  dom  F  =  O )
63, 5syl5sseq 3416 . . . 4  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( `' F " { 1 } ) 
C_  O )
7 indf 26484 . . . 4  |-  ( ( O  e.  V  /\  ( `' F " { 1 } )  C_  O
)  ->  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 } )
86, 7syldan 470 . . 3  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 } )
9 ffn 5571 . . 3  |-  ( ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  ( `' F " { 1 } ) )  Fn  O )
108, 9syl 16 . 2  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' F " { 1 } ) )  Fn  O
)
11 simpr 461 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F : O --> { 0 ,  1 } )
1211ffvelrnda 5855 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
0 ,  1 } )
13 prcom 3965 . . . 4  |-  { 0 ,  1 }  =  { 1 ,  0 }
1412, 13syl6eleq 2533 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
1 ,  0 } )
158ffvelrnda 5855 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  e.  {
0 ,  1 } )
1615, 13syl6eleq 2533 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  e.  {
1 ,  0 } )
17 simpll 753 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  O  e.  V )
186adantr 465 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( `' F " { 1 } )  C_  O )
19 simpr 461 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  x  e.  O )
20 ind1a 26489 . . . . 5  |-  ( ( O  e.  V  /\  ( `' F " { 1 } )  C_  O  /\  x  e.  O
)  ->  ( (
( (𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1  <->  x  e.  ( `' F " { 1 } ) ) )
2117, 18, 19, 20syl3anc 1218 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
( (𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1  <->  x  e.  ( `' F " { 1 } ) ) )
22 fniniseg 5836 . . . . . 6  |-  ( F  Fn  O  ->  (
x  e.  ( `' F " { 1 } )  <->  ( x  e.  O  /\  ( F `  x )  =  1 ) ) )
232, 22syl 16 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( x  e.  ( `' F " { 1 } )  <-> 
( x  e.  O  /\  ( F `  x
)  =  1 ) ) )
2423baibd 900 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( x  e.  ( `' F " { 1 } )  <-> 
( F `  x
)  =  1 ) )
2521, 24bitr2d 254 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( ( F `  x )  =  1  <->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1 ) )
2614, 16, 25elpreq 25912 . 2  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  =  ( ( (𝟭 `  O
) `  ( `' F " { 1 } ) ) `  x
) )
272, 10, 26eqfnfvd 5812 1  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  =  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3340   {csn 3889   {cpr 3891   `'ccnv 4851   dom cdm 4852   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430   0cc0 9294   1c1 9295  𝟭cind 26479
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-i2m1 9362  ax-1ne0 9363  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-ind 26480
This theorem is referenced by:  indf1ofs  26494
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