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Theorem funi 5188
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 4766 . 2  |-  Rel  _I
2 relcnv 5004 . . . . 5  |-  Rel  `'  _I
3 coi2 5141 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 10 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5038 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2276 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3174 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 4648 . 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 3094    _I cid 4241   `'ccnv 4625    o. ccom 4630   Rel wrel 4631   Fun wfun 4632
This theorem is referenced by:  cnvresid  5225  fnresi  5264  fvi  5478  ssdomg  6840  idcatfun  25273  domidmor  25280  codidmor  25282  grphidmor  25284  tendo02  30106
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 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-fun 4648
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