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Theorem f1ovi 5512
Description: The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
Assertion
Ref Expression
f1ovi  |-  _I  : _V
-1-1-onto-> _V

Proof of Theorem f1ovi
StepHypRef Expression
1 f1oi 5511 . 2  |-  (  _I  |`  _V ) : _V -1-1-onto-> _V
2 reli 4813 . . . 4  |-  Rel  _I
3 dfrel3 5131 . . . 4  |-  ( Rel 
_I 
<->  (  _I  |`  _V )  =  _I  )
42, 3mpbi 199 . . 3  |-  (  _I  |`  _V )  =  _I
5 f1oeq1 5463 . . 3  |-  ( (  _I  |`  _V )  =  _I  ->  ( (  _I  |`  _V ) : _V -1-1-onto-> _V  <->  _I  : _V -1-1-onto-> _V ) )
64, 5ax-mp 8 . 2  |-  ( (  _I  |`  _V ) : _V -1-1-onto-> _V  <->  _I  : _V -1-1-onto-> _V )
71, 6mpbi 199 1  |-  _I  : _V
-1-1-onto-> _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   _Vcvv 2788    _I cid 4304    |` cres 4691   Rel wrel 4694   -1-1-onto->wf1o 5254
This theorem is referenced by:  ncanth  6295  ginvsn  21016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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