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Theorem funi 5820
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 5159 . 2 Rel I
2 relcnv 5409 . . . . 5 Rel I
3 coi2 5555 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 5442 . . . 4 I = I
64, 5eqtri 2631 . . 3 ( I ∘ I ) = I
76eqimssi 3621 . 2 ( I ∘ I ) ⊆ I
8 df-fun 5792 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 956 1 Fun I
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wss 3539   I cid 4938  ccnv 5027  ccom 5032  Rel wrel 5033  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-fun 5792
This theorem is referenced by:  cnvresid  5868  fnresi  5908  fvi  6150  resiexd  6363  ssdomg  7865  tendo02  34917  residfi  40187
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