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Theorem indpreima 28682
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 5737 . . 3  |-  ( F : O --> { 0 ,  1 }  ->  F  Fn  O )
21adantl 467 . 2  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  Fn  O
)
3 cnvimass 5199 . . . . 5  |-  ( `' F " { 1 } )  C_  dom  F
4 fdm 5741 . . . . . 6  |-  ( F : O --> { 0 ,  1 }  ->  dom 
F  =  O )
54adantl 467 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  dom  F  =  O )
63, 5syl5sseq 3509 . . . 4  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( `' F " { 1 } ) 
C_  O )
7 indf 28673 . . . 4  |-  ( ( O  e.  V  /\  ( `' F " { 1 } )  C_  O
)  ->  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 } )
86, 7syldan 472 . . 3  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 } )
9 ffn 5737 . . 3  |-  ( ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) : O --> { 0 ,  1 }  ->  ( (𝟭 `  O ) `  ( `' F " { 1 } ) )  Fn  O )
108, 9syl 17 . 2  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' F " { 1 } ) )  Fn  O
)
11 simpr 462 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F : O --> { 0 ,  1 } )
1211ffvelrnda 6028 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
0 ,  1 } )
13 prcom 4072 . . . 4  |-  { 0 ,  1 }  =  { 1 ,  0 }
1412, 13syl6eleq 2518 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  e.  {
1 ,  0 } )
158ffvelrnda 6028 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  e.  {
0 ,  1 } )
1615, 13syl6eleq 2518 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  e.  {
1 ,  0 } )
17 simpll 758 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  O  e.  V )
186adantr 466 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( `' F " { 1 } )  C_  O )
19 simpr 462 . . . . 5  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  x  e.  O )
20 ind1a 28678 . . . . 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 1264 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( (
( (𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1  <->  x  e.  ( `' F " { 1 } ) ) )
22 fniniseg 6009 . . . . . 6  |-  ( F  Fn  O  ->  (
x  e.  ( `' F " { 1 } )  <->  ( x  e.  O  /\  ( F `  x )  =  1 ) ) )
232, 22syl 17 . . . . 5  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  ( x  e.  ( `' F " { 1 } )  <-> 
( x  e.  O  /\  ( F `  x
)  =  1 ) ) )
2423baibd 917 . . . 4  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( x  e.  ( `' F " { 1 } )  <-> 
( F `  x
)  =  1 ) )
2521, 24bitr2d 257 . . 3  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( ( F `  x )  =  1  <->  ( (
(𝟭 `  O ) `  ( `' F " { 1 } ) ) `  x )  =  1 ) )
2614, 16, 25elpreq 27992 . 2  |-  ( ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  /\  x  e.  O
)  ->  ( F `  x )  =  ( ( (𝟭 `  O
) `  ( `' F " { 1 } ) ) `  x
) )
272, 10, 26eqfnfvd 5985 1  |-  ( ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  =  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    C_ wss 3433   {csn 3993   {cpr 3995   `'ccnv 4844   dom cdm 4845   "cima 4848    Fn wfn 5587   -->wf 5588   ` cfv 5592   0cc0 9528   1c1 9529  𝟭cind 28668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-i2m1 9596  ax-1ne0 9597  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-ind 28669
This theorem is referenced by:  indf1ofs  28683
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