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Theorem funi 5142
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 4720 . 2  |-  Rel  _I
2 relcnv 4958 . . . . 5  |-  Rel  `'  _I
3 coi2 5095 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 10 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 4992 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2273 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3153 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 4602 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 891 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1619    C_ wss 3078    _I cid 4197   `'ccnv 4579    o. ccom 4584   Rel wrel 4585   Fun wfun 4586
This theorem is referenced by:  cnvresid  5179  fnresi  5218  fvi  5431  ssdomg  6793  idcatfun  25107  domidmor  25114  codidmor  25116  grphidmor  25118  tendo02  29735
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-fun 4602
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