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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 4812 . 2  |-  Rel  _I
2 relcnv 5050 . . . . 5  |-  Rel  `'  _I
3 coi2 5187 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5084 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2304 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3233 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5223 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 886 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1623    C_ wss 3153    _I cid 4303   `'ccnv 4687    o. ccom 4692   Rel wrel 4693   Fun wfun 5215
This theorem is referenced by:  cnvresid  5288  fnresi  5327  fvi  5541  ssdomg  6903  idcatfun  25352  domidmor  25359  codidmor  25361  grphidmor  25363  tendo02  30255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-fun 5223
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