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Theorem nvof1o 6169
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 5661 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
2 fdmrn 5731 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2sylib 198 . . . . 5  |-  ( F  Fn  A  ->  F : dom  F --> ran  F
)
43adantr 465 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : dom  F --> ran  F )
5 fndm 5663 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 465 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  dom  F  =  A )
7 df-rn 4836 . . . . . . 7  |-  ran  F  =  dom  `' F
8 dmeq 5026 . . . . . . 7  |-  ( `' F  =  F  ->  dom  `' F  =  dom  F )
97, 8syl5eq 2457 . . . . . 6  |-  ( `' F  =  F  ->  ran  F  =  dom  F
)
109, 5sylan9eqr 2467 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ran  F  =  A )
116, 10feq23d 5711 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( F : dom  F --> ran  F  <->  F : A --> A ) )
124, 11mpbid 212 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
--> A )
131adantr 465 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  F )
14 funeq 5590 . . . . 5  |-  ( `' F  =  F  -> 
( Fun  `' F  <->  Fun 
F ) )
1514adantl 466 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( Fun  `' F  <->  Fun  F ) )
1613, 15mpbird 234 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  `' F
)
17 df-f1 5576 . . 3  |-  ( F : A -1-1-> A  <->  ( F : A --> A  /\  Fun  `' F ) )
1812, 16, 17sylanbrc 664 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -1-1-> A )
19 simpl 457 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F  Fn  A )
20 df-fo 5577 . . 3  |-  ( F : A -onto-> A  <->  ( F  Fn  A  /\  ran  F  =  A ) )
2119, 10, 20sylanbrc 664 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -onto-> A )
22 df-f1o 5578 . 2  |-  ( F : A -1-1-onto-> A  <->  ( F : A -1-1-> A  /\  F : A -onto-> A ) )
2318, 21, 22sylanbrc 664 1  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
-1-1-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407   `'ccnv 4824   dom cdm 4825   ran crn 4826   Fun wfun 5565    Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   -onto->wfo 5569   -1-1-onto->wf1o 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578
This theorem is referenced by:  mirf1o  24438  lmif1o  24556
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