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Theorem funi 6530
Description: The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6626. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
funi Fun I

Proof of Theorem funi
StepHypRef Expression
1 reli 5780 . 2 Rel I
2 relcnv 6054 . . . . 5 Rel I
3 coi2 6213 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 6092 . . . 4 I = I
64, 5eqtri 2765 . . 3 ( I ∘ I ) = I
76eqimssi 4000 . 2 ( I ∘ I ) ⊆ I
8 df-fun 6495 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 709 1 Fun I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wss 3908   I cid 5528  ccnv 5630  ccom 5635  Rel wrel 5636  Fun wfun 6487
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6495
This theorem is referenced by:  cnvresid  6577  idfn  6626  fvi  6914  resiexd  7162  ssdomg  8898  residfi  9235  bj-funidres  35560  tendo02  39188
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