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Theorem funi 5476
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 4995 . 2  |-  Rel  _I
2 relcnv 5235 . . . . 5  |-  Rel  `'  _I
3 coi2 5379 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5269 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2456 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3395 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5449 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 887 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1652    C_ wss 3313    _I cid 4486   `'ccnv 4870    o. ccom 4875   Rel wrel 4876   Fun wfun 5441
This theorem is referenced by:  cnvresid  5516  fnresi  5555  fvi  5776  ssdomg  7146  tendo02  31522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-fun 5449
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