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Theorem funi 5223
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 4801 . 2  |-  Rel  _I
2 relcnv 5039 . . . . 5  |-  Rel  `'  _I
3 coi2 5176 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 10 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5073 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2278 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3207 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 4683 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 891 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1619    C_ wss 3127    _I cid 4276   `'ccnv 4660    o. ccom 4665   Rel wrel 4666   Fun wfun 4667
This theorem is referenced by:  cnvresid  5260  fnresi  5299  fvi  5513  ssdomg  6875  idcatfun  25309  domidmor  25316  codidmor  25318  grphidmor  25320  tendo02  30226
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-fun 4683
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