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Theorem funi 5387
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 4916 . 2  |-  Rel  _I
2 relcnv 5154 . . . . 5  |-  Rel  `'  _I
3 coi2 5292 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 8 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 5188 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2386 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3318 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5360 . 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 1647    C_ wss 3238    _I cid 4407   `'ccnv 4791    o. ccom 4796   Rel wrel 4797   Fun wfun 5352
This theorem is referenced by:  cnvresid  5427  fnresi  5466  fvi  5686  ssdomg  7050  tendo02  31035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-fun 5360
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