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Theorem funi 5423
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 4942 . 2  |-  Rel  _I
2 relcnv 5182 . . . . 5  |-  Rel  `'  _I
3 coi2 5326 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5216 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2407 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3345 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5396 . 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 1649    C_ wss 3263    _I cid 4434   `'ccnv 4817    o. ccom 4822   Rel wrel 4823   Fun wfun 5388
This theorem is referenced by:  cnvresid  5463  fnresi  5502  fvi  5722  ssdomg  7089  tendo02  30901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-fun 5396
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