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Theorem nvof1o 23052
Description: An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Assertion
Ref Expression
nvof1o  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
-1-1-onto-> A )

Proof of Theorem nvof1o
StepHypRef Expression
1 fnfun 5357 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
2 fdmrn 23051 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2sylib 188 . . . . 5  |-  ( F  Fn  A  ->  F : dom  F --> ran  F
)
43adantr 451 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : dom  F --> ran  F )
5 fndm 5359 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 451 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  dom  F  =  A )
7 df-rn 4716 . . . . . . 7  |-  ran  F  =  dom  `' F
8 dmeq 4895 . . . . . . 7  |-  ( `' F  =  F  ->  dom  `' F  =  dom  F )
97, 8syl5eq 2340 . . . . . 6  |-  ( `' F  =  F  ->  ran  F  =  dom  F
)
109, 5sylan9eqr 2350 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ran  F  =  A )
11 feq23 5394 . . . . 5  |-  ( ( dom  F  =  A  /\  ran  F  =  A )  ->  ( F : dom  F --> ran  F  <->  F : A --> A ) )
126, 10, 11syl2anc 642 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( F : dom  F --> ran  F  <->  F : A --> A ) )
134, 12mpbid 201 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
--> A )
144, 2sylibr 203 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  F )
15 funeq 5290 . . . . 5  |-  ( `' F  =  F  -> 
( Fun  `' F  <->  Fun 
F ) )
1615adantl 452 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( Fun  `' F  <->  Fun  F ) )
1714, 16mpbird 223 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  `' F
)
18 df-f1 5276 . . 3  |-  ( F : A -1-1-> A  <->  ( F : A --> A  /\  Fun  `' F ) )
1913, 17, 18sylanbrc 645 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -1-1-> A )
20 simpl 443 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F  Fn  A )
21 df-fo 5277 . . 3  |-  ( F : A -onto-> A  <->  ( F  Fn  A  /\  ran  F  =  A ) )
2220, 10, 21sylanbrc 645 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -onto-> A )
23 df-f1o 5278 . 2  |-  ( F : A -1-1-onto-> A  <->  ( F : A -1-1-> A  /\  F : A -onto-> A ) )
2419, 22, 23sylanbrc 645 1  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
-1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632   `'ccnv 4704   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  ballotlem7  23110
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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