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Theorem indf1ofs 28840
Description: The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
Assertion
Ref Expression
indf1ofs  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  Fin ) : ( ~P O  i^i  Fin ) -1-1-onto-> {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
Distinct variable group:    f, O
Allowed substitution hint:    V( f)

Proof of Theorem indf1ofs
Dummy variables  a 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 indf1o 28838 . . . 4  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
2 f1of1 5811 . . . 4  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O ) : ~P O -1-1-> ( { 0 ,  1 }  ^m  O ) )
31, 2syl 17 . . 3  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O -1-1-> ( { 0 ,  1 }  ^m  O ) )
4 inss1 3651 . . 3  |-  ( ~P O  i^i  Fin )  C_ 
~P O
5 f1ores 5826 . . 3  |-  ( ( (𝟭 `  O ) : ~P O -1-1-> ( { 0 ,  1 }  ^m  O )  /\  ( ~P O  i^i  Fin )  C_  ~P O )  ->  ( (𝟭 `  O
)  |`  ( ~P O  i^i  Fin ) ) : ( ~P O  i^i  Fin ) -1-1-onto-> ( (𝟭 `  O
) " ( ~P O  i^i  Fin )
) )
63, 4, 5sylancl 667 . 2  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) ) : ( ~P O  i^i  Fin )
-1-1-onto-> ( (𝟭 `  O ) " ( ~P O  i^i  Fin ) ) )
7 resres 5116 . . . 4  |-  ( ( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )
8 f1ofn 5813 . . . . . 6  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O )  Fn  ~P O )
9 fnresdm 5683 . . . . . 6  |-  ( (𝟭 `  O )  Fn  ~P O  ->  ( (𝟭 `  O
)  |`  ~P O )  =  (𝟭 `  O
) )
101, 8, 93syl 18 . . . . 5  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ~P O )  =  (𝟭 `  O ) )
1110reseq1d 5103 . . . 4  |-  ( O  e.  V  ->  (
( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  Fin ) )
127, 11syl5eqr 2498 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )  =  ( (𝟭 `  O )  |` 
Fin ) )
13 eqidd 2451 . . 3  |-  ( O  e.  V  ->  ( ~P O  i^i  Fin )  =  ( ~P O  i^i  Fin ) )
14 simpll 759 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  O  e.  V
)
15 simpr 463 . . . . . . . . . . . . . . 15  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ( ~P O  i^i  Fin ) )
164, 15sseldi 3429 . . . . . . . . . . . . . 14  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ~P O )
1716elpwid 3960 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  C_  O
)
18 indf 28830 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( (𝟭 `  O ) `  a ) : O --> { 0 ,  1 } )
1917, 18syldan 473 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
2019adantr 467 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
21 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a )  =  g )
2221feq1d 5712 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( ( (𝟭 `  O ) `  a
) : O --> { 0 ,  1 }  <->  g : O
--> { 0 ,  1 } ) )
2320, 22mpbid 214 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  g : O --> { 0 ,  1 } )
24 prex 4641 . . . . . . . . . . . 12  |-  { 0 ,  1 }  e.  _V
25 elmapg 7482 . . . . . . . . . . . 12  |-  ( ( { 0 ,  1 }  e.  _V  /\  O  e.  V )  ->  ( g  e.  ( { 0 ,  1 }  ^m  O )  <-> 
g : O --> { 0 ,  1 } ) )
2624, 25mpan 675 . . . . . . . . . . 11  |-  ( O  e.  V  ->  (
g  e.  ( { 0 ,  1 }  ^m  O )  <->  g : O
--> { 0 ,  1 } ) )
2726biimpar 488 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2814, 23, 27syl2anc 666 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2921cnveqd 5009 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  `' ( (𝟭 `  O ) `  a
)  =  `' g )
3029imaeq1d 5166 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  ( `' g " { 1 } ) )
31 indpi1 28836 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( `' ( (𝟭 `  O ) `  a
) " { 1 } )  =  a )
3217, 31syldan 473 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  a )
33 inss2 3652 . . . . . . . . . . . . 13  |-  ( ~P O  i^i  Fin )  C_ 
Fin
3433, 15sseldi 3429 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  Fin )
3532, 34eqeltrd 2528 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3635adantr 467 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3730, 36eqeltrrd 2529 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' g
" { 1 } )  e.  Fin )
3828, 37jca 535 . . . . . . . 8  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( g  e.  ( { 0 ,  1 }  ^m  O
)  /\  ( `' g " { 1 } )  e.  Fin )
)
3938ex 436 . . . . . . 7  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( ( (𝟭 `  O ) `  a
)  =  g  -> 
( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
) )
4039rexlimdva 2878 . . . . . 6  |-  ( O  e.  V  ->  ( E. a  e.  ( ~P O  i^i  Fin )
( (𝟭 `  O ) `  a )  =  g  ->  ( g  e.  ( { 0 ,  1 }  ^m  O
)  /\  ( `' g " { 1 } )  e.  Fin )
) )
41 cnvimass 5187 . . . . . . . . . 10  |-  ( `' g " { 1 } )  C_  dom  g
4226biimpa 487 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  g : O --> { 0 ,  1 } )
43 fdm 5731 . . . . . . . . . . . 12  |-  ( g : O --> { 0 ,  1 }  ->  dom  g  =  O )
4442, 43syl 17 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  dom  g  =  O )
4544adantrr 722 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  dom  g  =  O )
4641, 45syl5sseq 3479 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  C_  O )
47 simprr 765 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  Fin )
48 elfpw 7873 . . . . . . . . 9  |-  ( ( `' g " {
1 } )  e.  ( ~P O  i^i  Fin )  <->  ( ( `' g " { 1 } )  C_  O  /\  ( `' g " { 1 } )  e.  Fin ) )
4946, 47, 48sylanbrc 669 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  ( ~P O  i^i  Fin )
)
50 indpreima 28839 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  =  ( (𝟭 `  O ) `  ( `' g " { 1 } ) ) )
5150eqcomd 2456 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5242, 51syldan 473 . . . . . . . . 9  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5352adantrr 722 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( (𝟭 `  O ) `  ( `' g " {
1 } ) )  =  g )
54 fveq2 5863 . . . . . . . . . 10  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (𝟭 `  O ) `  a
)  =  ( (𝟭 `  O ) `  ( `' g " {
1 } ) ) )
5554eqeq1d 2452 . . . . . . . . 9  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (
(𝟭 `  O ) `  a )  =  g  <-> 
( (𝟭 `  O ) `  ( `' g " { 1 } ) )  =  g ) )
5655rspcev 3149 . . . . . . . 8  |-  ( ( ( `' g " { 1 } )  e.  ( ~P O  i^i  Fin )  /\  (
(𝟭 `  O ) `  ( `' g " {
1 } ) )  =  g )  ->  E. a  e.  ( ~P O  i^i  Fin )
( (𝟭 `  O ) `  a )  =  g )
5749, 53, 56syl2anc 666 . . . . . . 7  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  E. a  e.  ( ~P O  i^i  Fin ) ( (𝟭 `  O
) `  a )  =  g )
5857ex 436 . . . . . 6  |-  ( O  e.  V  ->  (
( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )  ->  E. a  e.  ( ~P O  i^i  Fin ) ( (𝟭 `  O
) `  a )  =  g ) )
5940, 58impbid 194 . . . . 5  |-  ( O  e.  V  ->  ( E. a  e.  ( ~P O  i^i  Fin )
( (𝟭 `  O ) `  a )  =  g  <-> 
( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
) )
601, 8syl 17 . . . . . 6  |-  ( O  e.  V  ->  (𝟭 `  O )  Fn  ~P O )
61 fvelimab 5919 . . . . . 6  |-  ( ( (𝟭 `  O )  Fn  ~P O  /\  ( ~P O  i^i  Fin )  C_ 
~P O )  -> 
( g  e.  ( (𝟭 `  O ) " ( ~P O  i^i  Fin ) )  <->  E. a  e.  ( ~P O  i^i  Fin ) ( (𝟭 `  O
) `  a )  =  g ) )
6260, 4, 61sylancl 667 . . . . 5  |-  ( O  e.  V  ->  (
g  e.  ( (𝟭 `  O ) " ( ~P O  i^i  Fin )
)  <->  E. a  e.  ( ~P O  i^i  Fin ) ( (𝟭 `  O
) `  a )  =  g ) )
63 cnveq 5007 . . . . . . . . 9  |-  ( f  =  g  ->  `' f  =  `' g
)
6463imaeq1d 5166 . . . . . . . 8  |-  ( f  =  g  ->  ( `' f " {
1 } )  =  ( `' g " { 1 } ) )
6564eleq1d 2512 . . . . . . 7  |-  ( f  =  g  ->  (
( `' f " { 1 } )  e.  Fin  <->  ( `' g " { 1 } )  e.  Fin )
)
6665elrab 3195 . . . . . 6  |-  ( g  e.  { f  e.  ( { 0 ,  1 }  ^m  O
)  |  ( `' f " { 1 } )  e.  Fin }  <-> 
( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)
6766a1i 11 . . . . 5  |-  ( O  e.  V  ->  (
g  e.  { f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " {
1 } )  e. 
Fin }  <->  ( g  e.  ( { 0 ,  1 }  ^m  O
)  /\  ( `' g " { 1 } )  e.  Fin )
) )
6859, 62, 673bitr4d 289 . . . 4  |-  ( O  e.  V  ->  (
g  e.  ( (𝟭 `  O ) " ( ~P O  i^i  Fin )
)  <->  g  e.  {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } ) )
6968eqrdv 2448 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O ) "
( ~P O  i^i  Fin ) )  =  {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
7012, 13, 69f1oeq123d 5809 . 2  |-  ( O  e.  V  ->  (
( (𝟭 `  O )  |`  ( ~P O  i^i  Fin ) ) : ( ~P O  i^i  Fin )
-1-1-onto-> ( (𝟭 `  O ) " ( ~P O  i^i  Fin ) )  <->  ( (𝟭 `  O )  |`  Fin ) : ( ~P O  i^i  Fin ) -1-1-onto-> { f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f
" { 1 } )  e.  Fin }
) )
716, 70mpbid 214 1  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  Fin ) : ( ~P O  i^i  Fin ) -1-1-onto-> {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   E.wrex 2737   {crab 2740   _Vcvv 3044    i^i cin 3402    C_ wss 3403   ~Pcpw 3950   {csn 3967   {cpr 3969   `'ccnv 4832   dom cdm 4833    |` cres 4835   "cima 4836    Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6288    ^m cmap 7469   Fincfn 7566   0cc0 9536   1c1 9537  𝟭cind 28825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-i2m1 9604  ax-1ne0 9605  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-map 7471  df-ind 28826
This theorem is referenced by:  eulerpartgbij  29198
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