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Theorem funi 6600
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6697. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
funi Fun I

Proof of Theorem funi
StepHypRef Expression
1 reli 5839 . 2 Rel I
2 relcnv 6125 . . . . 5 Rel I
3 coi2 6285 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 6164 . . . 4 I = I
64, 5eqtri 2763 . . 3 ( I ∘ I ) = I
76eqimssi 4056 . 2 ( I ∘ I ) ⊆ I
8 df-fun 6565 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 711 1 Fun I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wss 3963   I cid 5582  ccnv 5688  ccom 5693  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  cnvresid  6647  idfn  6697  fvi  6985  resiexd  7236  ssdomg  9039  residfi  9376  bj-funidres  37134  tendo02  40770  grimidvtxedg  47814
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