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Theorem nvof1o 6167
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 5671 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
2 fdmrn 5739 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2sylib 196 . . . . 5  |-  ( F  Fn  A  ->  F : dom  F --> ran  F
)
43adantr 465 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : dom  F --> ran  F )
5 fndm 5673 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 465 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  dom  F  =  A )
7 df-rn 5005 . . . . . . 7  |-  ran  F  =  dom  `' F
8 dmeq 5196 . . . . . . 7  |-  ( `' F  =  F  ->  dom  `' F  =  dom  F )
97, 8syl5eq 2515 . . . . . 6  |-  ( `' F  =  F  ->  ran  F  =  dom  F
)
109, 5sylan9eqr 2525 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ran  F  =  A )
116, 10feq23d 5719 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( F : dom  F --> ran  F  <->  F : A --> A ) )
124, 11mpbid 210 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
--> A )
131adantr 465 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  F )
14 funeq 5600 . . . . 5  |-  ( `' F  =  F  -> 
( Fun  `' F  <->  Fun 
F ) )
1514adantl 466 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( Fun  `' F  <->  Fun  F ) )
1613, 15mpbird 232 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  `' F
)
17 df-f1 5586 . . 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 5587 . . 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 5588 . 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 184    /\ wa 369    = wceq 1374   `'ccnv 4993   dom cdm 4994   ran crn 4995   Fun wfun 5575    Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588
This theorem is referenced by:  mirf1o  23757  lmif1o  23832
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