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Theorem indf1ofs 26451
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 26449 . . . 4  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
2 f1of1 5635 . . . 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 3565 . . 3  |-  ( ~P O  i^i  Fin )  C_ 
~P O
5 f1ores 5650 . . 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 5118 . . . 4  |-  ( ( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )
8 f1ofn 5637 . . . . . 6  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O )  Fn  ~P O )
9 fnresdm 5515 . . . . . 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 5104 . . . 4  |-  ( O  e.  V  ->  (
( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  Fin ) )
127, 11syl5eqr 2484 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )  =  ( (𝟭 `  O )  |` 
Fin ) )
13 eqidd 2439 . . 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 3349 . . . . . . . . . . . . . 14  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ~P O )
1716elpwid 3865 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  C_  O
)
18 indf 26441 . . . . . . . . . . . . 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 5541 . . . . . . . . . . 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 4529 . . . . . . . . . . . 12  |-  { 0 ,  1 }  e.  _V
25 elmapg 7219 . . . . . . . . . . . 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 5010 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  `' ( (𝟭 `  O ) `  a
)  =  `' g )
3029imaeq1d 5163 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  ( `' g " { 1 } ) )
31 indpi1 26447 . . . . . . . . . . . . 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 3566 . . . . . . . . . . . . 13  |-  ( ~P O  i^i  Fin )  C_ 
Fin
3433, 15sseldi 3349 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  Fin )
3532, 34eqeltrd 2512 . . . . . . . . . . 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 2513 . . . . . . . . 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 2836 . . . . . 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 5184 . . . . . . . . . 10  |-  ( `' g " { 1 } )  C_  dom  g
4226biimpa 484 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  g : O --> { 0 ,  1 } )
43 fdm 5558 . . . . . . . . . . . 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 3399 . . . . . . . . 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 7605 . . . . . . . . 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 26450 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  =  ( (𝟭 `  O ) `  ( `' g " { 1 } ) ) )
5150eqcomd 2443 . . . . . . . . . 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 5686 . . . . . . . . . 10  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (𝟭 `  O ) `  a
)  =  ( (𝟭 `  O ) `  ( `' g " {
1 } ) ) )
5554eqeq1d 2446 . . . . . . . . 9  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (
(𝟭 `  O ) `  a )  =  g  <-> 
( (𝟭 `  O ) `  ( `' g " { 1 } ) )  =  g ) )
5655rspcev 3068 . . . . . . . 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 5742 . . . . . 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 5008 . . . . . . . . 9  |-  ( f  =  g  ->  `' f  =  `' g
)
6463imaeq1d 5163 . . . . . . . 8  |-  ( f  =  g  ->  ( `' f " {
1 } )  =  ( `' g " { 1 } ) )
6564eleq1d 2504 . . . . . . 7  |-  ( f  =  g  ->  (
( `' f " { 1 } )  e.  Fin  <->  ( `' g " { 1 } )  e.  Fin )
)
6665elrab 3112 . . . . . 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 2436 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O ) "
( ~P O  i^i  Fin ) )  =  {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
7012, 13, 69f1oeq123d 5633 . 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 1369    e. wcel 1756   E.wrex 2711   {crab 2714   _Vcvv 2967    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   {csn 3872   {cpr 3874   `'ccnv 4834   dom cdm 4835    |` cres 4837   "cima 4838    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    ^m cmap 7206   Fincfn 7302   0cc0 9274   1c1 9275  𝟭cind 26436
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-i2m1 9342  ax-1ne0 9343  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-ind 26437
This theorem is referenced by:  eulerpartgbij  26724
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