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Theorem funi 5150
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 4663 . 2  |-  Rel  _I
2 relcnv 4912 . . . . 5  |-  Rel  `'  _I
3 coi2 5050 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 5 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 4938 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2158 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3148 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5120 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 926 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1331    C_ wss 3066    _I cid 4205   `'ccnv 4533    o. ccom 4538   Rel wrel 4539   Fun wfun 5112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120
This theorem is referenced by:  cnvresid  5192  fnresi  5235  fvi  5471  ssdomg  6665  climshft2  11068
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