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Theorem f1finf1o 6803
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 109 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
2 simplr 504 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  B  e.  Fin )
3 f1rn 5299 . . . . . 6  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
43adantl 275 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  C_  B )
5 f1fn 5300 . . . . . . . . 9  |-  ( F : A -1-1-> B  ->  F  Fn  A )
6 fnima 5211 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 14 . . . . . . . 8  |-  ( F : A -1-1-> B  -> 
( F " A
)  =  ran  F
)
87adantl 275 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F " A )  =  ran  F )
9 ssid 3087 . . . . . . . . 9  |-  A  C_  A
109a1i 9 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  C_  A
)
11 simpll 503 . . . . . . . . 9  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  ~~  B )
12 enfii 6736 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  A  ~~  B )  ->  A  e.  Fin )
132, 11, 12syl2anc 408 . . . . . . . 8  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  A  e.  Fin )
14 f1imaeng 6654 . . . . . . . 8  |-  ( ( F : A -1-1-> B  /\  A  C_  A  /\  A  e.  Fin )  ->  ( F " A
)  ~~  A )
151, 10, 13, 14syl3anc 1201 . . . . . . 7  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ( F " A )  ~~  A
)
168, 15eqbrtrrd 3922 . . . . . 6  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  ~~  A )
17 entr 6646 . . . . . 6  |-  ( ( ran  F  ~~  A  /\  A  ~~  B )  ->  ran  F  ~~  B )
1816, 11, 17syl2anc 408 . . . . 5  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  ~~  B )
19 fisseneq 6788 . . . . 5  |-  ( ( B  e.  Fin  /\  ran  F  C_  B  /\  ran  F  ~~  B )  ->  ran  F  =  B )
202, 4, 18, 19syl3anc 1201 . . . 4  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  ran  F  =  B )
21 dff1o5 5344 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
221, 20, 21sylanbrc 413 . . 3  |-  ( ( ( A  ~~  B  /\  B  e.  Fin )  /\  F : A -1-1-> B )  ->  F : A
-1-1-onto-> B )
2322ex 114 . 2  |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  ->  F : A -1-1-onto-> B
) )
24 f1of1 5334 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
2523, 24impbid1 141 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 103    <-> wb 104    = wceq 1316    e. wcel 1465    C_ wss 3041   class class class wbr 3899   ran crn 4510   "cima 4512    Fn wfn 5088   -1-1->wf1 5090   -1-1-onto->wf1o 5092    ~~ cen 6600   Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1o 6281  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by:  iseqf1olemqf1o  10234  crth  11827  pwf1oexmid  13121
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