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Theorem indf1ofs 26626
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 26624 . . . 4  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
2 f1of1 5747 . . . 4  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O ) : ~P O -1-1-> ( { 0 ,  1 }  ^m  O ) )
31, 2syl 16 . . 3  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O -1-1-> ( { 0 ,  1 }  ^m  O ) )
4 inss1 3677 . . 3  |-  ( ~P O  i^i  Fin )  C_ 
~P O
5 f1ores 5762 . . 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 662 . 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 5230 . . . 4  |-  ( ( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )
8 f1ofn 5749 . . . . . 6  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O )  Fn  ~P O )
9 fnresdm 5627 . . . . . 6  |-  ( (𝟭 `  O )  Fn  ~P O  ->  ( (𝟭 `  O
)  |`  ~P O )  =  (𝟭 `  O
) )
101, 8, 93syl 20 . . . . 5  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ~P O )  =  (𝟭 `  O ) )
1110reseq1d 5216 . . . 4  |-  ( O  e.  V  ->  (
( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  Fin ) )
127, 11syl5eqr 2509 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )  =  ( (𝟭 `  O )  |` 
Fin ) )
13 eqidd 2455 . . 3  |-  ( O  e.  V  ->  ( ~P O  i^i  Fin )  =  ( ~P O  i^i  Fin ) )
14 simpll 753 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  O  e.  V
)
15 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ( ~P O  i^i  Fin ) )
164, 15sseldi 3461 . . . . . . . . . . . . . 14  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ~P O )
1716elpwid 3977 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  C_  O
)
18 indf 26616 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( (𝟭 `  O ) `  a ) : O --> { 0 ,  1 } )
1917, 18syldan 470 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
2019adantr 465 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
21 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a )  =  g )
2221feq1d 5653 . . . . . . . . . . 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 210 . . . . . . . . . 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 7336 . . . . . . . . . . . 12  |-  ( ( { 0 ,  1 }  e.  _V  /\  O  e.  V )  ->  ( g  e.  ( { 0 ,  1 }  ^m  O )  <-> 
g : O --> { 0 ,  1 } ) )
2624, 25mpan 670 . . . . . . . . . . 11  |-  ( O  e.  V  ->  (
g  e.  ( { 0 ,  1 }  ^m  O )  <->  g : O
--> { 0 ,  1 } ) )
2726biimpar 485 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2814, 23, 27syl2anc 661 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2921cnveqd 5122 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  `' ( (𝟭 `  O ) `  a
)  =  `' g )
3029imaeq1d 5275 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  ( `' g " { 1 } ) )
31 indpi1 26622 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( `' ( (𝟭 `  O ) `  a
) " { 1 } )  =  a )
3217, 31syldan 470 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  a )
33 inss2 3678 . . . . . . . . . . . . 13  |-  ( ~P O  i^i  Fin )  C_ 
Fin
3433, 15sseldi 3461 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  Fin )
3532, 34eqeltrd 2542 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3635adantr 465 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3730, 36eqeltrrd 2543 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' g
" { 1 } )  e.  Fin )
3828, 37jca 532 . . . . . . . 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 434 . . . . . . 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 2945 . . . . . 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 5296 . . . . . . . . . 10  |-  ( `' g " { 1 } )  C_  dom  g
4226biimpa 484 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  g : O --> { 0 ,  1 } )
43 fdm 5670 . . . . . . . . . . . 12  |-  ( g : O --> { 0 ,  1 }  ->  dom  g  =  O )
4442, 43syl 16 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  dom  g  =  O )
4544adantrr 716 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  dom  g  =  O )
4641, 45syl5sseq 3511 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  C_  O )
47 simprr 756 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  Fin )
48 elfpw 7723 . . . . . . . . 9  |-  ( ( `' g " {
1 } )  e.  ( ~P O  i^i  Fin )  <->  ( ( `' g " { 1 } )  C_  O  /\  ( `' g " { 1 } )  e.  Fin ) )
4946, 47, 48sylanbrc 664 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  ( ~P O  i^i  Fin )
)
50 indpreima 26625 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  =  ( (𝟭 `  O ) `  ( `' g " { 1 } ) ) )
5150eqcomd 2462 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5242, 51syldan 470 . . . . . . . . 9  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5352adantrr 716 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( (𝟭 `  O ) `  ( `' g " {
1 } ) )  =  g )
54 fveq2 5798 . . . . . . . . . 10  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (𝟭 `  O ) `  a
)  =  ( (𝟭 `  O ) `  ( `' g " {
1 } ) ) )
5554eqeq1d 2456 . . . . . . . . 9  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (
(𝟭 `  O ) `  a )  =  g  <-> 
( (𝟭 `  O ) `  ( `' g " { 1 } ) )  =  g ) )
5655rspcev 3177 . . . . . . . 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 661 . . . . . . 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 434 . . . . . 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 191 . . . . 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 16 . . . . . 6  |-  ( O  e.  V  ->  (𝟭 `  O )  Fn  ~P O )
61 fvelimab 5855 . . . . . 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 662 . . . . 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 5120 . . . . . . . . 9  |-  ( f  =  g  ->  `' f  =  `' g
)
6463imaeq1d 5275 . . . . . . . 8  |-  ( f  =  g  ->  ( `' f " {
1 } )  =  ( `' g " { 1 } ) )
6564eleq1d 2523 . . . . . . 7  |-  ( f  =  g  ->  (
( `' f " { 1 } )  e.  Fin  <->  ( `' g " { 1 } )  e.  Fin )
)
6665elrab 3222 . . . . . 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 285 . . . 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 2451 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O ) "
( ~P O  i^i  Fin ) )  =  {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
7012, 13, 69f1oeq123d 5745 . 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 210 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 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2799   {crab 2802   _Vcvv 3076    i^i cin 3434    C_ wss 3435   ~Pcpw 3967   {csn 3984   {cpr 3986   `'ccnv 4946   dom cdm 4947    |` cres 4949   "cima 4950    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199    ^m cmap 7323   Fincfn 7419   0cc0 9392   1c1 9393  𝟭cind 26611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-i2m1 9460  ax-1ne0 9461  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-map 7325  df-ind 26612
This theorem is referenced by:  eulerpartgbij  26898
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