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Theorem funi 6534
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6630. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
funi Fun I

Proof of Theorem funi
StepHypRef Expression
1 reli 5783 . 2 Rel I
2 relcnv 6057 . . . . 5 Rel I
3 coi2 6216 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 6095 . . . 4 I = I
64, 5eqtri 2761 . . 3 ( I ∘ I ) = I
76eqimssi 4003 . 2 ( I ∘ I ) ⊆ I
8 df-fun 6499 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 710 1 Fun I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wss 3911   I cid 5531  ccnv 5633  ccom 5638  Rel wrel 5639  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-fun 6499
This theorem is referenced by:  cnvresid  6581  idfn  6630  fvi  6918  resiexd  7167  ssdomg  8943  residfi  9280  bj-funidres  35668  tendo02  39296
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