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Theorem funi 5286
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 4815 . 2  |-  Rel  _I
2 relcnv 5053 . . . . 5  |-  Rel  `'  _I
3 coi2 5191 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5087 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2305 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3234 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5259 . 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 1625    C_ wss 3154    _I cid 4306   `'ccnv 4690    o. ccom 4695   Rel wrel 4696   Fun wfun 5251
This theorem is referenced by:  cnvresid  5324  fnresi  5363  fvi  5581  ssdomg  6909  idcatfun  25952  domidmor  25959  codidmor  25961  grphidmor  25963  tendo02  31049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-fun 5259
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