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Theorem fihashf1rn 10535
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 5330 . . 3  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 simpl 108 . . 3  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  A  e.  Fin )
3 fnfi 6825 . . 3  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
41, 2, 3syl2an2 583 . 2  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  F  e.  Fin )
5 f1o2ndf1 6125 . . . 4  |-  ( F : A -1-1-> B  -> 
( 2nd  |`  F ) : F -1-1-onto-> ran  F )
6 df-2nd 6039 . . . . . . . . 9  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
76funmpt2 5162 . . . . . . . 8  |-  Fun  2nd
8 f1f 5328 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F : A --> B )
98anim2i 339 . . . . . . . . . 10  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( A  e. 
Fin  /\  F : A
--> B ) )
109ancomd 265 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( F : A
--> B  /\  A  e. 
Fin ) )
11 fex 5647 . . . . . . . . 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 5641 . . . . . . . 8  |-  ( ( Fun  2nd  /\  F  e. 
_V )  ->  ( 2nd  |`  F )  e. 
_V )
147, 12, 13sylancr 410 . . . . . . 7  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( 2nd  |`  F )  e.  _V )
15 f1oeq1 5356 . . . . . . . . . 10  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  <->  f : F
-1-1-onto-> ran  F ) )
1615biimpd 143 . . . . . . . . 9  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
1716eqcoms 2142 . . . . . . . 8  |-  ( f  =  ( 2nd  |`  F )  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
1817adantl 275 . . . . . . 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 2772 . . . . . 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 114 . . . . 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 80 . . . 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 124 . 2  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  E. f  f : F -1-1-onto-> ran  F )
24 fihasheqf1oi 10534 . . . 4  |-  ( ( F  e.  Fin  /\  f : F -1-1-onto-> ran  F )  -> 
( `  F )  =  ( `  ran  F ) )
2524ex 114 . . 3  |-  ( F  e.  Fin  ->  (
f : F -1-1-onto-> ran  F  ->  ( `  F )  =  ( `  ran  F ) ) )
2625exlimdv 1791 . 2  |-  ( F  e.  Fin  ->  ( E. f  f : F
-1-1-onto-> ran  F  ->  ( `  F
)  =  ( `  ran  F ) ) )
274, 23, 26sylc 62 1  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686   {csn 3527   U.cuni 3736   ran crn 4540    |` cres 4541   Fun wfun 5117    Fn wfn 5118   -->wf 5119   -1-1->wf1 5120   -1-1-onto->wf1o 5122   ` cfv 5123   2ndc2nd 6037   Fincfn 6634  ♯chash 10521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-ihash 10522
This theorem is referenced by: (None)
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