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Theorem funi 5284
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
funi  |-  Fun  _I

Proof of Theorem funi
StepHypRef Expression
1 reli 4813 . 2  |-  Rel  _I
2 relcnv 5051 . . . . 5  |-  Rel  `'  _I
3 coi2 5189 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5085 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2303 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3232 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5257 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 886 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1623    C_ wss 3152    _I cid 4304   `'ccnv 4688    o. ccom 4693   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  cnvresid  5322  fnresi  5361  fvi  5579  ssdomg  6907  idcatfun  25941  domidmor  25948  codidmor  25950  grphidmor  25952  tendo02  30976
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-fun 5257
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