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Theorem funi 5205
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 4718 . 2 Rel I
2 relcnv 4967 . . . . 5 Rel I
3 coi2 5105 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 4993 . . . 4 I = I
64, 5eqtri 2178 . . 3 ( I ∘ I ) = I
76eqimssi 3184 . 2 ( I ∘ I ) ⊆ I
8 df-fun 5175 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 927 1 Fun I
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wss 3102   I cid 4251  ccnv 4588  ccom 4593  Rel wrel 4594  Fun wfun 5167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-fun 5175
This theorem is referenced by:  cnvresid  5247  fnresi  5290  fvi  5528  ssdomg  6726  climshft2  11215
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