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Theorem f1oi 5527
Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
f1oi  |-  (  _I  |`  A ) : A -1-1-onto-> A

Proof of Theorem f1oi
StepHypRef Expression
1 fnresi 5377 . 2  |-  (  _I  |`  A )  Fn  A
2 cnvresid 5338 . . . 4  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
32fneq1i 5354 . . 3  |-  ( `' (  _I  |`  A )  Fn  A  <->  (  _I  |`  A )  Fn  A
)
41, 3mpbir 200 . 2  |-  `' (  _I  |`  A )  Fn  A
5 dff1o4 5496 . 2  |-  ( (  _I  |`  A ) : A -1-1-onto-> A  <->  ( (  _I  |`  A )  Fn  A  /\  `' (  _I  |`  A )  Fn  A ) )
61, 4, 5mpbir2an 886 1  |-  (  _I  |`  A ) : A -1-1-onto-> A
Colors of variables: wff set class
Syntax hints:    _I cid 4320   `'ccnv 4704    |` cres 4707    Fn wfn 5266   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1ovi  5528  fveqf1o  5822  isoid  5842  enrefg  6909  ssdomg  6923  hartogslem1  7273  wdomref  7302  infxpenc  7661  pwfseqlem5  8301  dfle2  10497  wunndx  13180  idfucl  13771  idffth  13823  ressffth  13828  setccatid  13932  idghm  14714  symggrp  14796  symgid  14797  ssidcn  17001  resthauslem  17107  sshauslem  17116  hausdiag  17355  idqtop  17413  fmid  17671  mbfid  19007  dvid  19283  dvexp  19318  wilthlem2  20323  wilthlem3  20324  hoif  22350  idunop  22574  idcnop  22577  elunop2  22609  subfacp1lem4  23729  subfacp1lem5  23730  ghomsn  24010  scprefat  25174  dispos  25390  idfisf  25944  idsubfun  25961  infemb  25962  idcatfun  26044  mzpresrename  26931  eldioph2lem1  26942  eldioph2lem2  26943  diophren  26999  kelac2  27266  islinds2  27386  lindfres  27396  lindsmm  27401  lnrfg  27426  idlaut  30907  tendoidcl  31580  tendo0co2  31599  erng1r  31806  dvalveclem  31837  dva0g  31839  dvh0g  31923
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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