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Theorem f1finf1o 7086
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 447 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
2 f1f 5437 . . . . . . 7  |-  ( F : A -1-1-> B  ->  F : A --> B )
32adantl 452 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
--> B )
4 ffn 5389 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
53, 4syl 15 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F  Fn  A )
6 simpll 730 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  B )
7 frn 5395 . . . . . . . . . 10  |-  ( F : A --> B  ->  ran  F  C_  B )
83, 7syl 15 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  C_  B )
9 df-pss 3168 . . . . . . . . . 10  |-  ( ran 
F  C.  B  <->  ( ran  F 
C_  B  /\  ran  F  =/=  B ) )
109baib 871 . . . . . . . . 9  |-  ( ran 
F  C_  B  ->  ( ran  F  C.  B  <->  ran 
F  =/=  B ) )
118, 10syl 15 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  <->  ran  F  =/= 
B ) )
12 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  B  e.  Fin )
13 relen 6868 . . . . . . . . . . . . . . 15  |-  Rel  ~~
1413brrelexi 4729 . . . . . . . . . . . . . 14  |-  ( A 
~~  B  ->  A  e.  _V )
156, 14syl 15 . . . . . . . . . . . . 13  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  e.  _V )
16 elmapg 6785 . . . . . . . . . . . . 13  |-  ( ( B  e.  Fin  /\  A  e.  _V )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
1712, 15, 16syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F  e.  ( B  ^m  A
)  <->  F : A --> B ) )
183, 17mpbird 223 . . . . . . . . . . 11  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F  e.  ( B  ^m  A ) )
19 f1f1orn 5483 . . . . . . . . . . . 12  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2019adantl 452 . . . . . . . . . . 11  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> ran  F )
21 f1oen3g 6877 . . . . . . . . . . 11  |-  ( ( F  e.  ( B  ^m  A )  /\  F : A -1-1-onto-> ran  F )  ->  A  ~~  ran  F )
2218, 20, 21syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  ran  F )
23 php3 7047 . . . . . . . . . . . 12  |-  ( ( B  e.  Fin  /\  ran  F  C.  B )  ->  ran  F  ~<  B )
2423ex 423 . . . . . . . . . . 11  |-  ( B  e.  Fin  ->  ( ran  F  C.  B  ->  ran  F  ~<  B )
)
2512, 24syl 15 . . . . . . . . . 10  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  ran  F 
~<  B ) )
26 ensdomtr 6997 . . . . . . . . . 10  |-  ( ( A  ~~  ran  F  /\  ran  F  ~<  B )  ->  A  ~<  B )
2722, 25, 26ee12an 1353 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  A  ~<  B ) )
28 sdomnen 6890 . . . . . . . . 9  |-  ( A 
~<  B  ->  -.  A  ~~  B )
2927, 28syl6 29 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F 
C.  B  ->  -.  A  ~~  B ) )
3011, 29sylbird 226 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( ran  F  =/=  B  ->  -.  A  ~~  B ) )
3130necon4ad 2507 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( A  ~~  B  ->  ran  F  =  B ) )
326, 31mpd 14 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  =  B )
33 df-fo 5261 . . . . 5  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
345, 32, 33sylanbrc 645 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -onto-> B )
35 df-f1o 5262 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
361, 34, 35sylanbrc 645 . . 3  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> B )
3736ex 423 . 2  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )
38 f1of1 5471 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
3937, 38impbid1 194 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152    C. wpss 3153   class class class wbr 4023   ran crn 4690    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254  (class class class)co 5858    ^m cmap 6772    ~~ cen 6860    ~< csdm 6862   Fincfn 6863
This theorem is referenced by:  hashfac  11396  crt  12846  eulerthlem2  12850  fidomndrnglem  16047  basellem4  20321  lgsqrlem4  20583  lgseisenlem2  20589  enf1f1oOLD  26397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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