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Theorem funi 4956
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 4487 . 2  |-  Rel  _I
2 relcnv 4727 . . . . 5  |-  Rel  `'  _I
3 coi2 4861 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 7 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 4752 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2102 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3054 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 4928 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 884 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1285    C_ wss 2974    _I cid 4045   `'ccnv 4364    o. ccom 4369   Rel wrel 4370   Fun wfun 4920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-fun 4928
This theorem is referenced by:  cnvresid  4998  fnresi  5041  fvi  5256  ssdomg  6317  climshft2  10272
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