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

Theorem f1finf1o 7335
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 5639 . . . . . . 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 5591 . . . . . 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 5597 . . . . . . . . . 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 3336 . . . . . . . . . 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 7114 . . . . . . . . . . . . . . 15  |-  Rel  ~~
1413brrelexi 4918 . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . 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 5685 . . . . . . . . . . . 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 7123 . . . . . . . . . . 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 7293 . . . . . . . . . . . 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 7243 . . . . . . . . . 10  |-  ( ( A  ~~  ran  F  /\  ran  F  ~<  B )  ->  A  ~<  B )
2722, 25, 26ee12an 1372 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  A  ~<  B ) )
28 sdomnen 7136 . . . . . . . . 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 2665 . . . . . 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 5460 . . . . 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 5461 . . . 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 5673 . 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 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    C_ wss 3320    C. wpss 3321   class class class wbr 4212   ran crn 4879    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453  (class class class)co 6081    ^m cmap 7018    ~~ cen 7106    ~< csdm 7108   Fincfn 7109
This theorem is referenced by:  hashfac  11707  crt  13167  eulerthlem2  13171  fidomndrnglem  16366  basellem4  20866  lgsqrlem4  21128  lgseisenlem2  21134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113
  Copyright terms: Public domain W3C validator