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Theorem f1finf1o 7102
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 5453 . . . . . . 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 5405 . . . . . 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 5411 . . . . . . . . . 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 3181 . . . . . . . . . 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 6884 . . . . . . . . . . . . . . 15  |-  Rel  ~~
1413brrelexi 4745 . . . . . . . . . . . . . 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 6801 . . . . . . . . . . . . 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 5499 . . . . . . . . . . . 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 6893 . . . . . . . . . . 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 7063 . . . . . . . . . . . 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 7013 . . . . . . . . . 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 6906 . . . . . . . . 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 2520 . . . . . 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 5277 . . . . 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 5278 . . . 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 5487 . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165    C. wpss 3166   class class class wbr 4039   ran crn 4706    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270  (class class class)co 5874    ^m cmap 6788    ~~ cen 6876    ~< csdm 6878   Fincfn 6879
This theorem is referenced by:  hashfac  11412  crt  12862  eulerthlem2  12866  fidomndrnglem  16063  basellem4  20337  lgsqrlem4  20599  lgseisenlem2  20605  enf1f1oOLD  26500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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