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Theorem indpreima 27789
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 5731 . . 3  |-  ( F : O --> { 0 ,  1 }  ->  F  Fn  O )
21adantl 466 . 2  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  Fn  O
)
3 cnvimass 5357 . . . . 5  |-  ( `' F " { 1 } )  C_  dom  F
4 fdm 5735 . . . . . 6  |-  ( F : O --> { 0 ,  1 }  ->  dom 
F  =  O )
54adantl 466 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  dom  F  =  O )
63, 5syl5sseq 3552 . . . 4  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( `' F " { 1 } ) 
C_  O )
7 indf 27780 . . . 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 5731 . . 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 6022 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
0 ,  1 } )
13 prcom 4105 . . . 4  |-  { 0 ,  1 }  =  { 1 ,  0 }
1412, 13syl6eleq 2565 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
1 ,  0 } )
158ffvelrnda 6022 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  e.  {
0 ,  1 } )
1615, 13syl6eleq 2565 . . 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 27785 . . . . 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 1228 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
( (𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1  <->  x  e.  ( `' F " { 1 } ) ) )
22 fniniseg 6003 . . . . . 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 907 . . . 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 27188 . 2  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  =  ( ( (𝟭 `  O
) `  ( `' F " { 1 } ) ) `  x
) )
272, 10, 26eqfnfvd 5979 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 1379    e. wcel 1767    C_ wss 3476   {csn 4027   {cpr 4029   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588   0cc0 9493   1c1 9494  𝟭cind 27775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-i2m1 9561  ax-1ne0 9562  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-ind 27776
This theorem is referenced by:  indf1ofs  27790
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