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Theorem funi 5155
 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 4668 . 2 Rel I
2 relcnv 4917 . . . . 5 Rel I
3 coi2 5055 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 5 . . . 4 ( I ∘ I ) = I
5 cnvi 4943 . . . 4 I = I
64, 5eqtri 2160 . . 3 ( I ∘ I ) = I
76eqimssi 3153 . 2 ( I ∘ I ) ⊆ I
8 df-fun 5125 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 926 1 Fun I
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ⊆ wss 3071   I cid 4210  ◡ccnv 4538   ∘ ccom 4543  Rel wrel 4544  Fun wfun 5117 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125 This theorem is referenced by:  cnvresid  5197  fnresi  5240  fvi  5478  ssdomg  6672  climshft2  11075
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