Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nvof1o Structured version   Unicode version

Theorem nvof1o 24032
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 5534 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
2 fdmrn 24031 . . . . . 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 5536 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 452 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  dom  F  =  A )
7 df-rn 4881 . . . . . . 7  |-  ran  F  =  dom  `' F
8 dmeq 5062 . . . . . . 7  |-  ( `' F  =  F  ->  dom  `' F  =  dom  F )
97, 8syl5eq 2479 . . . . . 6  |-  ( `' F  =  F  ->  ran  F  =  dom  F
)
109, 5sylan9eqr 2489 . . . . 5  |-  ( ( F  Fn  A  /\  `' F  =  F
)  ->  ran  F  =  A )
116, 10feq23d 5580 . . . 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 5465 . . . . 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 5451 . . 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 5452 . . 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 5453 . 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 1652   `'ccnv 4869   dom cdm 4870   ran crn 4871   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
  Copyright terms: Public domain W3C validator