MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1finf1o Unicode version

Theorem f1finf1o 7294
Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
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 448 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
2 f1f 5598 . . . . . . 7  |-  ( F : A -1-1-> B  ->  F : A --> B )
32adantl 453 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
--> B )
4 ffn 5550 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
53, 4syl 16 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F  Fn  A )
6 simpll 731 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  B )
7 frn 5556 . . . . . . . . . 10  |-  ( F : A --> B  ->  ran  F  C_  B )
83, 7syl 16 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  C_  B )
9 df-pss 3296 . . . . . . . . . 10  |-  ( ran 
F  C.  B  <->  ( ran  F 
C_  B  /\  ran  F  =/=  B ) )
109baib 872 . . . . . . . . 9  |-  ( ran 
F  C_  B  ->  ( ran  F  C.  B  <->  ran 
F  =/=  B ) )
118, 10syl 16 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  <->  ran  F  =/= 
B ) )
12 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  B  e.  Fin )
13 relen 7073 . . . . . . . . . . . . . . 15  |-  Rel  ~~
1413brrelexi 4877 . . . . . . . . . . . . . 14  |-  ( A 
~~  B  ->  A  e.  _V )
156, 14syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  e.  _V )
16 elmapg 6990 . . . . . . . . . . . . 13  |-  ( ( B  e.  Fin  /\  A  e.  _V )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
1712, 15, 16syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F  e.  ( B  ^m  A
)  <->  F : A --> B ) )
183, 17mpbird 224 . . . . . . . . . . 11  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F  e.  ( B  ^m  A ) )
19 f1f1orn 5644 . . . . . . . . . . . 12  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2019adantl 453 . . . . . . . . . . 11  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> ran  F )
21 f1oen3g 7082 . . . . . . . . . . 11  |-  ( ( F  e.  ( B  ^m  A )  /\  F : A -1-1-onto-> ran  F )  ->  A  ~~  ran  F )
2218, 20, 21syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  ran  F )
23 php3 7252 . . . . . . . . . . . 12  |-  ( ( B  e.  Fin  /\  ran  F  C.  B )  ->  ran  F  ~<  B )
2423ex 424 . . . . . . . . . . 11  |-  ( B  e.  Fin  ->  ( ran  F  C.  B  ->  ran  F  ~<  B )
)
2512, 24syl 16 . . . . . . . . . 10  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  ran  F 
~<  B ) )
26 ensdomtr 7202 . . . . . . . . . 10  |-  ( ( A  ~~  ran  F  /\  ran  F  ~<  B )  ->  A  ~<  B )
2722, 25, 26ee12an 1369 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  A  ~<  B ) )
28 sdomnen 7095 . . . . . . . . 9  |-  ( A 
~<  B  ->  -.  A  ~~  B )
2927, 28syl6 31 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  -.  A  ~~  B ) )
3011, 29sylbird 227 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F  =/=  B  ->  -.  A  ~~  B ) )
3130necon4ad 2628 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( A  ~~  B  ->  ran  F  =  B ) )
326, 31mpd 15 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  =  B )
33 df-fo 5419 . . . . 5  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
345, 32, 33sylanbrc 646 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -onto-> B )
35 df-f1o 5420 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
361, 34, 35sylanbrc 646 . . 3  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> B )
3736ex 424 . 2  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )
38 f1of1 5632 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
3937, 38impbid1 195 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:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    C_ wss 3280    C. wpss 3281   class class class wbr 4172   ran crn 4838    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412  (class class class)co 6040    ^m cmap 6977    ~~ cen 7065    ~< csdm 7067   Fincfn 7068
This theorem is referenced by:  hashfac  11662  crt  13122  eulerthlem2  13126  fidomndrnglem  16321  basellem4  20819  lgsqrlem4  21081  lgseisenlem2  21087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072
  Copyright terms: Public domain W3C validator