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Theorem nvof1o 23876
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 5475 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
2 fdmrn 23875 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2sylib 189 . . . . 5  |-  ( F  Fn  A  ->  F : dom  F --> ran  F
)
43adantr 452 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : dom  F --> ran  F )
5 fndm 5477 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 452 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  dom  F  =  A )
7 df-rn 4822 . . . . . . 7  |-  ran  F  =  dom  `' F
8 dmeq 5003 . . . . . . 7  |-  ( `' F  =  F  ->  dom  `' F  =  dom  F )
97, 8syl5eq 2424 . . . . . 6  |-  ( `' F  =  F  ->  ran  F  =  dom  F
)
109, 5sylan9eqr 2434 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ran  F  =  A )
116, 10feq23d 5521 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( F : dom  F --> ran  F  <->  F : A --> A ) )
124, 11mpbid 202 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
--> A )
131adantr 452 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  F )
14 funeq 5406 . . . . 5  |-  ( `' F  =  F  -> 
( Fun  `' F  <->  Fun 
F ) )
1514adantl 453 . . . 4  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ( Fun  `' F  <->  Fun  F ) )
1613, 15mpbird 224 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  Fun  `' F
)
17 df-f1 5392 . . 3  |-  ( F : A -1-1-> A  <->  ( F : A --> A  /\  Fun  `' F ) )
1812, 16, 17sylanbrc 646 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -1-1-> A )
19 simpl 444 . . 3  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F  Fn  A )
20 df-fo 5393 . . 3  |-  ( F : A -onto-> A  <->  ( F  Fn  A  /\  ran  F  =  A ) )
2119, 10, 20sylanbrc 646 . 2  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A -onto-> A )
22 df-f1o 5394 . 2  |-  ( F : A -1-1-onto-> A  <->  ( F : A -1-1-> A  /\  F : A -onto-> A ) )
2318, 21, 22sylanbrc 646 1  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  F : A
-1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   `'ccnv 4810   dom cdm 4811   ran crn 4812   Fun wfun 5381    Fn wfn 5382   -->wf 5383   -1-1->wf1 5384   -onto->wfo 5385   -1-1-onto->wf1o 5386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394
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