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Theorem funi 5543
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 5060 . 2  |-  Rel  _I
2 relcnv 5304 . . . . 5  |-  Rel  `'  _I
3 coi2 5449 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 5 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5337 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2425 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3488 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5515 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 918 1  |-  Fun  _I
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    C_ wss 3406    _I cid 4721   `'ccnv 4929    o. ccom 4934   Rel wrel 4935   Fun wfun 5507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-opab 4443  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-fun 5515
This theorem is referenced by:  cnvresid  5583  fnresi  5623  fvi  5848  resiexd  6059  ssdomg  7502  residfi  32674  usgresvm1  32800  usgresvm1ALT  32804  tendo02  36965
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