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Theorem funi 4932
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 4465 . 2 Rel I
2 relcnv 4703 . . . . 5 Rel I
3 coi2 4837 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 7 . . . 4 ( I ∘ I ) = I
5 cnvi 4728 . . . 4 I = I
64, 5eqtri 2060 . . 3 ( I ∘ I ) = I
76eqimssi 2999 . 2 ( I ∘ I ) ⊆ I
8 df-fun 4904 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 849 1 Fun I
Colors of variables: wff set class
Syntax hints:   = wceq 1243  wss 2917   I cid 4025  ccnv 4344  ccom 4349  Rel wrel 4350  Fun wfun 4896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904
This theorem is referenced by:  cnvresid  4973  fnresi  5016  fvi  5230  ssdomg  6258  climshft2  9827
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