Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  indf1ofs Structured version   Unicode version

Theorem indf1ofs 24424
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 24422 . . . 4  |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
2 f1of1 5674 . . . 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 3562 . . 3  |-  ( ~P O  i^i  Fin )  C_ 
~P O
5 f1ores 5690 . . 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 645 . 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 5160 . . . 4  |-  ( ( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )
8 f1ofn 5676 . . . . . 6  |-  ( (𝟭 `  O ) : ~P O
-1-1-onto-> ( { 0 ,  1 }  ^m  O )  ->  (𝟭 `  O )  Fn  ~P O )
9 fnresdm 5555 . . . . . 6  |-  ( (𝟭 `  O )  Fn  ~P O  ->  ( (𝟭 `  O
)  |`  ~P O )  =  (𝟭 `  O
) )
101, 8, 93syl 19 . . . . 5  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ~P O )  =  (𝟭 `  O ) )
1110reseq1d 5146 . . . 4  |-  ( O  e.  V  ->  (
( (𝟭 `  O )  |` 
~P O )  |`  Fin )  =  (
(𝟭 `  O )  |`  Fin ) )
127, 11syl5eqr 2483 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O )  |`  ( ~P O  i^i  Fin ) )  =  ( (𝟭 `  O )  |` 
Fin ) )
13 eqidd 2438 . . 3  |-  ( O  e.  V  ->  ( ~P O  i^i  Fin )  =  ( ~P O  i^i  Fin ) )
14 simpll 732 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  O  e.  V
)
15 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ( ~P O  i^i  Fin ) )
164, 15sseldi 3347 . . . . . . . . . . . . . 14  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  ~P O )
1716elpwid 3809 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  C_  O
)
18 indf 24414 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( (𝟭 `  O ) `  a ) : O --> { 0 ,  1 } )
1917, 18syldan 458 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
2019adantr 453 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a ) : O --> { 0 ,  1 } )
21 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( (𝟭 `  O
) `  a )  =  g )
2221feq1d 5581 . . . . . . . . . . 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 203 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  g : O --> { 0 ,  1 } )
24 prex 4407 . . . . . . . . . . . 12  |-  { 0 ,  1 }  e.  _V
25 elmapg 7032 . . . . . . . . . . . 12  |-  ( ( { 0 ,  1 }  e.  _V  /\  O  e.  V )  ->  ( g  e.  ( { 0 ,  1 }  ^m  O )  <-> 
g : O --> { 0 ,  1 } ) )
2624, 25mpan 653 . . . . . . . . . . 11  |-  ( O  e.  V  ->  (
g  e.  ( { 0 ,  1 }  ^m  O )  <->  g : O
--> { 0 ,  1 } ) )
2726biimpar 473 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2814, 23, 27syl2anc 644 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  g  e.  ( { 0 ,  1 }  ^m  O ) )
2921cnveqd 5049 . . . . . . . . . . 11  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  `' ( (𝟭 `  O ) `  a
)  =  `' g )
3029imaeq1d 5203 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  ( `' g " { 1 } ) )
31 indpi1 24420 . . . . . . . . . . . . 13  |-  ( ( O  e.  V  /\  a  C_  O )  -> 
( `' ( (𝟭 `  O ) `  a
) " { 1 } )  =  a )
3217, 31syldan 458 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  =  a )
33 inss2 3563 . . . . . . . . . . . . 13  |-  ( ~P O  i^i  Fin )  C_ 
Fin
3433, 15sseldi 3347 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  a  e.  Fin )
3532, 34eqeltrd 2511 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin ) )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3635adantr 453 . . . . . . . . . 10  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' ( (𝟭 `  O ) `  a ) " {
1 } )  e. 
Fin )
3730, 36eqeltrrd 2512 . . . . . . . . 9  |-  ( ( ( O  e.  V  /\  a  e.  ( ~P O  i^i  Fin )
)  /\  ( (𝟭 `  O ) `  a
)  =  g )  ->  ( `' g
" { 1 } )  e.  Fin )
3828, 37jca 520 . . . . . . . 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 425 . . . . . . 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 2831 . . . . . 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 5225 . . . . . . . . . 10  |-  ( `' g " { 1 } )  C_  dom  g
4226biimpa 472 . . . . . . . . . . . 12  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  g : O --> { 0 ,  1 } )
43 fdm 5596 . . . . . . . . . . . 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 699 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  dom  g  =  O )
4641, 45syl5sseq 3397 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  C_  O )
47 simprr 735 . . . . . . . . 9  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  Fin )
48 elfpw 7409 . . . . . . . . 9  |-  ( ( `' g " {
1 } )  e.  ( ~P O  i^i  Fin )  <->  ( ( `' g " { 1 } )  C_  O  /\  ( `' g " { 1 } )  e.  Fin ) )
4946, 47, 48sylanbrc 647 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( `' g " { 1 } )  e.  ( ~P O  i^i  Fin )
)
50 indpreima 24423 . . . . . . . . . . 11  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  g  =  ( (𝟭 `  O ) `  ( `' g " { 1 } ) ) )
5150eqcomd 2442 . . . . . . . . . 10  |-  ( ( O  e.  V  /\  g : O --> { 0 ,  1 } )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5242, 51syldan 458 . . . . . . . . 9  |-  ( ( O  e.  V  /\  g  e.  ( {
0 ,  1 }  ^m  O ) )  ->  ( (𝟭 `  O
) `  ( `' g " { 1 } ) )  =  g )
5352adantrr 699 . . . . . . . 8  |-  ( ( O  e.  V  /\  ( g  e.  ( { 0 ,  1 }  ^m  O )  /\  ( `' g
" { 1 } )  e.  Fin )
)  ->  ( (𝟭 `  O ) `  ( `' g " {
1 } ) )  =  g )
54 fveq2 5729 . . . . . . . . . 10  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (𝟭 `  O ) `  a
)  =  ( (𝟭 `  O ) `  ( `' g " {
1 } ) ) )
5554eqeq1d 2445 . . . . . . . . 9  |-  ( a  =  ( `' g
" { 1 } )  ->  ( (
(𝟭 `  O ) `  a )  =  g  <-> 
( (𝟭 `  O ) `  ( `' g " { 1 } ) )  =  g ) )
5655rspcev 3053 . . . . . . . 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 644 . . . . . . 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 425 . . . . . 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 185 . . . . 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 5783 . . . . . 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 645 . . . . 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 5047 . . . . . . . . 9  |-  ( f  =  g  ->  `' f  =  `' g
)
6463imaeq1d 5203 . . . . . . . 8  |-  ( f  =  g  ->  ( `' f " {
1 } )  =  ( `' g " { 1 } ) )
6564eleq1d 2503 . . . . . . 7  |-  ( f  =  g  ->  (
( `' f " { 1 } )  e.  Fin  <->  ( `' g " { 1 } )  e.  Fin )
)
6665elrab 3093 . . . . . 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 278 . . . 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 2435 . . 3  |-  ( O  e.  V  ->  (
(𝟭 `  O ) "
( ~P O  i^i  Fin ) )  =  {
f  e.  ( { 0 ,  1 }  ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
7012, 13, 69f1oeq123d 5672 . 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 203 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   {crab 2710   _Vcvv 2957    i^i cin 3320    C_ wss 3321   ~Pcpw 3800   {csn 3815   {cpr 3816   `'ccnv 4878   dom cdm 4879    |` cres 4881   "cima 4882    Fn wfn 5450   -->wf 5451   -1-1->wf1 5452   -1-1-onto->wf1o 5454   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   Fincfn 7110   0cc0 8991   1c1 8992  𝟭cind 24409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-i2m1 9059  ax-1ne0 9060  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-ind 24410
  Copyright terms: Public domain W3C validator