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Theorem funi 5209
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 4787 . 2  |-  Rel  _I
2 relcnv 5025 . . . . 5  |-  Rel  `'  _I
3 coi2 5162 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 10 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5059 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2276 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3193 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 4669 . 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 3113    _I cid 4262   `'ccnv 4646    o. ccom 4651   Rel wrel 4652   Fun wfun 4653
This theorem is referenced by:  cnvresid  5246  fnresi  5285  fvi  5499  ssdomg  6861  idcatfun  25294  domidmor  25301  codidmor  25303  grphidmor  25305  tendo02  30127
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 4101  ax-nul 4109  ax-pr 4172
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-fun 4669
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