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Theorem funi 5250
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 4758 . 2  |-  Rel  _I
2 relcnv 5008 . . . . 5  |-  Rel  `'  _I
3 coi2 5147 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 5 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5035 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2198 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3213 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5220 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 942 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1353    C_ wss 3131    _I cid 4290   `'ccnv 4627    o. ccom 4632   Rel wrel 4633   Fun wfun 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-fun 5220
This theorem is referenced by:  cnvresid  5292  fnresi  5335  fvi  5575  ssdomg  6780  climshft2  11316
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