ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funi Unicode version

Theorem funi 5032
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 4553 . 2  |-  Rel  _I
2 relcnv 4797 . . . . 5  |-  Rel  `'  _I
3 coi2 4934 . . . . 5  |-  ( Rel  `'  _I  ->  (  _I  o.  `'  _I  )  =  `'  _I  )
42, 3ax-mp 7 . . . 4  |-  (  _I  o.  `'  _I  )  =  `'  _I
5 cnvi 4823 . . . 4  |-  `'  _I  =  _I
64, 5eqtri 2108 . . 3  |-  (  _I  o.  `'  _I  )  =  _I
76eqimssi 3078 . 2  |-  (  _I  o.  `'  _I  )  C_  _I
8 df-fun 5004 . 2  |-  ( Fun 
_I 
<->  ( Rel  _I  /\  (  _I  o.  `'  _I  )  C_  _I  )
)
91, 7, 8mpbir2an 888 1  |-  Fun  _I
Colors of variables: wff set class
Syntax hints:    = wceq 1289    C_ wss 2997    _I cid 4106   `'ccnv 4427    o. ccom 4432   Rel wrel 4433   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-fun 5004
This theorem is referenced by:  cnvresid  5074  fnresi  5117  fvi  5345  ssdomg  6475  climshft2  10659
  Copyright terms: Public domain W3C validator