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Theorem funi 6366
 Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6455. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
funi Fun I

Proof of Theorem funi
StepHypRef Expression
1 reli 5675 . 2 Rel I
2 relcnv 5945 . . . . 5 Rel I
3 coi2 6094 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 5978 . . . 4 I = I
64, 5eqtri 2845 . . 3 ( I ∘ I ) = I
76eqimssi 4000 . 2 ( I ∘ I ) ⊆ I
8 df-fun 6336 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 710 1 Fun I
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ⊆ wss 3908   I cid 5436  ◡ccnv 5531   ∘ ccom 5536  Rel wrel 5537  Fun wfun 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-fun 6336 This theorem is referenced by:  cnvresid  6412  idfn  6455  fnresiOLD  6457  fvi  6722  resiexd  6961  ssdomg  8542  residfi  8793  bj-funidres  34527  tendo02  38041
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