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Theorem fihashf1rn 10046
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 5169 . . 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 6574 . . 3  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
41, 2, 3syl2an2 559 . 2  |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  F  e.  Fin )
5 f1o2ndf1 5931 . . . 4  |-  ( F : A -1-1-> B  -> 
( 2nd  |`  F ) : F -1-1-onto-> ran  F )
6 df-2nd 5850 . . . . . . . . 9  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
76funmpt2 5009 . . . . . . . 8  |-  Fun  2nd
8 f1f 5167 . . . . . . . . . . 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 5467 . . . . . . . . 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 5461 . . . . . . . 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 5195 . . . . . . . . . 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 2088 . . . . . . . 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 2697 . . . . . 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 10045 . . . 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 1744 . 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 1287   E.wex 1424    e. wcel 1436   _Vcvv 2614   {csn 3425   U.cuni 3630   ran crn 4405    |` cres 4406   Fun wfun 4966    Fn wfn 4967   -->wf 4968   -1-1->wf1 4969   -1-1-onto->wf1o 4971   ` cfv 4972   2ndc2nd 5848   Fincfn 6390  ♯chash 10032
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-2nd 5850  df-recs 6005  df-frec 6091  df-1o 6116  df-er 6225  df-en 6391  df-dom 6392  df-fin 6393  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-ihash 10033
This theorem is referenced by: (None)
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