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Theorem funi 5032
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 4553 . 2 Rel I
2 relcnv 4797 . . . . 5 Rel I
3 coi2 4934 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 7 . . . 4 ( I ∘ I ) = I
5 cnvi 4823 . . . 4 I = I
64, 5eqtri 2108 . . 3 ( I ∘ I ) = I
76eqimssi 3078 . 2 ( I ∘ I ) ⊆ I
8 df-fun 5004 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 888 1 Fun I
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wss 2997   I cid 4106  ccnv 4427  ccom 4432  Rel wrel 4433  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-fun 5004
This theorem is referenced by:  cnvresid  5074  fnresi  5117  fvi  5345  ssdomg  6475  climshft2  10659
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