ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fihashf1rn Unicode version

Theorem fihashf1rn 10185
Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
Assertion
Ref Expression
fihashf1rn  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )

Proof of Theorem fihashf1rn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 f1fn 5212 . . 3  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 simpl 107 . . 3  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  A  e.  Fin )
3 fnfi 6636 . . 3  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
41, 2, 3syl2an2 561 . 2  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  F  e.  Fin )
5 f1o2ndf1 5985 . . . 4  |-  ( F : A -1-1-> B  -> 
( 2nd  |`  F ) : F -1-1-onto-> ran  F )
6 df-2nd 5904 . . . . . . . . 9  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
76funmpt2 5047 . . . . . . . 8  |-  Fun  2nd
8 f1f 5210 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
98anim2i 334 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( A  e. 
Fin  /\  F : A
--> B ) )
109ancomd 263 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( F : A
--> B  /\  A  e. 
Fin ) )
11 fex 5516 . . . . . . . . 9  |-  ( ( F : A --> B  /\  A  e.  Fin )  ->  F  e.  _V )
1210, 11syl 14 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  F  e.  _V )
13 resfunexg 5510 . . . . . . . 8  |-  ( ( Fun  2nd  /\  F  e. 
_V )  ->  ( 2nd  |`  F )  e. 
_V )
147, 12, 13sylancr 405 . . . . . . 7  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( 2nd  |`  F )  e.  _V )
15 f1oeq1 5238 . . . . . . . . . 10  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  <->  f : F
-1-1-onto-> ran  F ) )
1615biimpd 142 . . . . . . . . 9  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
1716eqcoms 2091 . . . . . . . 8  |-  ( f  =  ( 2nd  |`  F )  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
1817adantl 271 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  F : A -1-1-> B
)  /\  f  =  ( 2nd  |`  F )
)  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
1914, 18spcimedv 2705 . . . . . 6  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  E. f  f : F -1-1-onto-> ran  F ) )
2019ex 113 . . . . 5  |-  ( A  e.  Fin  ->  ( F : A -1-1-> B  -> 
( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  E. f 
f : F -1-1-onto-> ran  F
) ) )
2120com13 79 . . . 4  |-  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  ( F : A -1-1-> B  -> 
( A  e.  Fin  ->  E. f  f : F -1-1-onto-> ran  F ) ) )
225, 21mpcom 36 . . 3  |-  ( F : A -1-1-> B  -> 
( A  e.  Fin  ->  E. f  f : F -1-1-onto-> ran  F ) )
2322impcom 123 . 2  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  E. f  f : F -1-1-onto-> ran  F )
24 fihasheqf1oi 10184 . . . 4  |-  ( ( F  e.  Fin  /\  f : F -1-1-onto-> ran  F )  -> 
( `  F )  =  ( `  ran  F ) )
2524ex 113 . . 3  |-  ( F  e.  Fin  ->  (
f : F -1-1-onto-> ran  F  ->  ( `  F )  =  ( `  ran  F ) ) )
2625exlimdv 1747 . 2  |-  ( F  e.  Fin  ->  ( E. f  f : F
-1-1-onto-> ran  F  ->  ( `  F
)  =  ( `  ran  F ) ) )
274, 23, 26sylc 61 1  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   {csn 3444   U.cuni 3651   ran crn 4437    |` cres 4438   Fun wfun 5004    Fn wfn 5005   -->wf 5006   -1-1->wf1 5007   -1-1-onto->wf1o 5009   ` cfv 5010   2ndc2nd 5902   Fincfn 6447  ♯chash 10171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-ltadd 7451
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-2nd 5904  df-recs 6062  df-frec 6148  df-1o 6173  df-er 6282  df-en 6448  df-dom 6449  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010  df-ihash 10172
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator