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Theorem hashf1rn 12389
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 Alexander van der Vekens, 12-Jan-2018.) (Revised by Alexander van der Vekens, 4-Feb-2018.)
Assertion
Ref Expression
hashf1rn  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A -1-1-> B  ->  ( # `  F
)  =  ( # `  ran  F ) ) )

Proof of Theorem hashf1rn
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 f1f 5779 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex 6131 . . . . . . . . 9  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
32ex 434 . . . . . . . 8  |-  ( F : A --> B  -> 
( A  e.  V  ->  F  e.  _V )
)
41, 3syl 16 . . . . . . 7  |-  ( F : A -1-1-> B  -> 
( A  e.  V  ->  F  e.  _V )
)
54com12 31 . . . . . 6  |-  ( A  e.  V  ->  ( F : A -1-1-> B  ->  F  e.  _V )
)
65adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A -1-1-> B  ->  F  e.  _V ) )
76imp 429 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  F  e.  _V )
8 rnexg 6713 . . . . 5  |-  ( F  e.  _V  ->  ran  F  e.  _V )
98ancli 551 . . . 4  |-  ( F  e.  _V  ->  ( F  e.  _V  /\  ran  F  e.  _V ) )
107, 9syl 16 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( F  e.  _V  /\ 
ran  F  e.  _V ) )
11 f1o2ndf1 6888 . . . . 5  |-  ( F : A -1-1-> B  -> 
( 2nd  |`  F ) : F -1-1-onto-> ran  F )
12 df-2nd 6782 . . . . . . . . . 10  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1312funmpt2 5623 . . . . . . . . 9  |-  Fun  2nd
142expcom 435 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( F : A --> B  ->  F  e.  _V )
)
1514adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A --> B  ->  F  e.  _V ) )
161, 15syl5com 30 . . . . . . . . . 10  |-  ( F : A -1-1-> B  -> 
( ( A  e.  V  /\  B  e.  W )  ->  F  e.  _V ) )
1716impcom 430 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  F  e.  _V )
18 resfunexg 6124 . . . . . . . . 9  |-  ( ( Fun  2nd  /\  F  e. 
_V )  ->  ( 2nd  |`  F )  e. 
_V )
1913, 17, 18sylancr 663 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( 2nd  |`  F )  e.  _V )
20 f1oeq1 5805 . . . . . . . . . . 11  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  <->  f : F
-1-1-onto-> ran  F ) )
2120biimpd 207 . . . . . . . . . 10  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
2221eqcoms 2479 . . . . . . . . 9  |-  ( f  =  ( 2nd  |`  F )  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
2322adantl 466 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  F : A -1-1-> B )  /\  f  =  ( 2nd  |`  F ) )  -> 
( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
2419, 23spcimedv 3197 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  E. f 
f : F -1-1-onto-> ran  F
) )
2524ex 434 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A -1-1-> B  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  E. f  f : F -1-1-onto-> ran  F ) ) )
2625com13 80 . . . . 5  |-  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  ( F : A -1-1-> B  -> 
( ( A  e.  V  /\  B  e.  W )  ->  E. f 
f : F -1-1-onto-> ran  F
) ) )
2711, 26mpcom 36 . . . 4  |-  ( F : A -1-1-> B  -> 
( ( A  e.  V  /\  B  e.  W )  ->  E. f 
f : F -1-1-onto-> ran  F
) )
2827impcom 430 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  E. f  f : F
-1-1-onto-> ran  F )
29 hasheqf1oi 12388 . . 3  |-  ( ( F  e.  _V  /\  ran  F  e.  _V )  ->  ( E. f  f : F -1-1-onto-> ran  F  ->  ( # `
 F )  =  ( # `  ran  F ) ) )
3010, 28, 29sylc 60 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( # `  F )  =  ( # `  ran  F ) )
3130ex 434 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A -1-1-> B  ->  ( # `  F
)  =  ( # `  ran  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113   {csn 4027   U.cuni 4245   ran crn 5000    |` cres 5001   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   -1-1-onto->wf1o 5585   ` cfv 5586   2ndc2nd 6780   #chash 12369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-hash 12370
This theorem is referenced by:  hashimarn  12458  sizeusglecusg  24162  usgsizedg  31864  usgsizedgALT  31865  usgsizedgALT2  31866
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