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Theorem hashf1rn 12470
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 5763 . . . . . . . 8  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex 6125 . . . . . . . . 9  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
32ex 432 . . . . . . . 8  |-  ( F : A --> B  -> 
( A  e.  V  ->  F  e.  _V )
)
41, 3syl 17 . . . . . . 7  |-  ( F : A -1-1-> B  -> 
( A  e.  V  ->  F  e.  _V )
)
54com12 29 . . . . . 6  |-  ( A  e.  V  ->  ( F : A -1-1-> B  ->  F  e.  _V )
)
65adantr 463 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A -1-1-> B  ->  F  e.  _V ) )
76imp 427 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  F  e.  _V )
8 rnexg 6715 . . . 4  |-  ( F  e.  _V  ->  ran  F  e.  _V )
97, 8jccir 537 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( F  e.  _V  /\ 
ran  F  e.  _V ) )
10 f1o2ndf1 6891 . . . . 5  |-  ( F : A -1-1-> B  -> 
( 2nd  |`  F ) : F -1-1-onto-> ran  F )
11 df-2nd 6784 . . . . . . . . . 10  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1211funmpt2 5605 . . . . . . . . 9  |-  Fun  2nd
132expcom 433 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( F : A --> B  ->  F  e.  _V )
)
1413adantr 463 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( F : A --> B  ->  F  e.  _V ) )
151, 14syl5com 28 . . . . . . . . . 10  |-  ( F : A -1-1-> B  -> 
( ( A  e.  V  /\  B  e.  W )  ->  F  e.  _V ) )
1615impcom 428 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  F  e.  _V )
17 resfunexg 6117 . . . . . . . . 9  |-  ( ( Fun  2nd  /\  F  e. 
_V )  ->  ( 2nd  |`  F )  e. 
_V )
1812, 16, 17sylancr 661 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( 2nd  |`  F )  e.  _V )
19 f1oeq1 5789 . . . . . . . . . . 11  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  <->  f : F
-1-1-onto-> ran  F ) )
2019biimpd 207 . . . . . . . . . 10  |-  ( ( 2nd  |`  F )  =  f  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
2120eqcoms 2414 . . . . . . . . 9  |-  ( f  =  ( 2nd  |`  F )  ->  ( ( 2nd  |`  F ) : F -1-1-onto-> ran  F  ->  f : F -1-1-onto-> ran  F ) )
2221adantl 464 . . . . . . . 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 ) )
2318, 22spcimedv 3142 . . . . . . 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
) )
2423ex 432 . . . . . 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 ) ) )
2524com13 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
) ) )
2610, 25mpcom 34 . . . 4  |-  ( F : A -1-1-> B  -> 
( ( A  e.  V  /\  B  e.  W )  ->  E. f 
f : F -1-1-onto-> ran  F
) )
2726impcom 428 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  ->  E. f  f : F
-1-1-onto-> ran  F )
28 hasheqf1oi 12469 . . 3  |-  ( ( F  e.  _V  /\  ran  F  e.  _V )  ->  ( E. f  f : F -1-1-onto-> ran  F  ->  ( # `
 F )  =  ( # `  ran  F ) ) )
299, 27, 28sylc 59 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  F : A -1-1-> B )  -> 
( # `  F )  =  ( # `  ran  F ) )
3029ex 432 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 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058   {csn 3971   U.cuni 4190   ran crn 4823    |` cres 4824   Fun wfun 5562   -->wf 5564   -1-1->wf1 5565   -1-1-onto->wf1o 5567   ` cfv 5568   2ndc2nd 6782   #chash 12450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-hash 12451
This theorem is referenced by:  hashimarn  12543  sizeusglecusg  24890  usgsizedg  38005  usgsizedgALT  38006  usgsizedgALT2  38007
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