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Theorem funi 4875
 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 4408 . 2 Rel I
2 relcnv 4646 . . . . 5 Rel I
3 coi2 4780 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 7 . . . 4 ( I ∘ I ) = I
5 cnvi 4671 . . . 4 I = I
64, 5eqtri 2057 . . 3 ( I ∘ I ) = I
76eqimssi 2993 . 2 ( I ∘ I ) ⊆ I
8 df-fun 4847 . 2 (Fun I ↔ (Rel I ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 848 1 Fun I
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ⊆ wss 2911   I cid 4016  ◡ccnv 4287   ∘ ccom 4292  Rel wrel 4293  Fun wfun 4839 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847 This theorem is referenced by:  cnvresid  4916  fnresi  4959  fvi  5173  ssdomg  6194
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