ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1finf1o Unicode version

Theorem f1finf1o 6580
Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
Assertion
Ref Expression
f1finf1o  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <-> 
F : A -1-1-onto-> B ) )

Proof of Theorem f1finf1o
StepHypRef Expression
1 simpr 108 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
2 simplr 497 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  B  e.  Fin )
3 f1rn 5165 . . . . . 6  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
43adantl 271 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  C_  B )
5 f1fn 5166 . . . . . . . . 9  |-  ( F : A -1-1-> B  ->  F  Fn  A )
6 fnima 5085 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 14 . . . . . . . 8  |-  ( F : A -1-1-> B  -> 
( F " A
)  =  ran  F
)
87adantl 271 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F " A )  =  ran  F )
9 ssid 3029 . . . . . . . . 9  |-  A  C_  A
109a1i 9 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  C_  A
)
11 simpll 496 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  B )
12 enfii 6520 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  ~~  B )  ->  A  e.  Fin )
132, 11, 12syl2anc 403 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  e.  Fin )
14 f1imaeng 6439 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  C_  A  /\  A  e.  Fin )  ->  ( F " A
)  ~~  A )
151, 10, 13, 14syl3anc 1170 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F " A )  ~~  A
)
168, 15eqbrtrrd 3833 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  ~~  A )
17 entr 6431 . . . . . 6  |-  ( ( ran  F  ~~  A  /\  A  ~~  B )  ->  ran  F  ~~  B )
1816, 11, 17syl2anc 403 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  ~~  B )
19 fisseneq 6567 . . . . 5  |-  ( ( B  e.  Fin  /\  ran  F  C_  B  /\  ran  F  ~~  B )  ->  ran  F  =  B )
202, 4, 18, 19syl3anc 1170 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  =  B )
21 dff1o5 5210 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
221, 20, 21sylanbrc 408 . . 3  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> B )
2322ex 113 . 2  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )
24 f1of1 5200 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
2523, 24impbid1 140 1  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <-> 
F : A -1-1-onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434    C_ wss 2984   class class class wbr 3811   ran crn 4402   "cima 4404    Fn wfn 4964   -1-1->wf1 4966   -1-1-onto->wf1o 4968    ~~ cen 6385   Fincfn 6387
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390
This theorem is referenced by:  crth  10980
  Copyright terms: Public domain W3C validator