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Theorem funi 4960
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 4493 . 2 Rel I
2 relcnv 4731 . . . . 5 Rel I
3 coi2 4865 . . . . 5 (Rel I → ( I ∘ I ) = I )
42, 3ax-mp 7 . . . 4 ( I ∘ I ) = I
5 cnvi 4756 . . . 4 I = I
64, 5eqtri 2076 . . 3 ( I ∘ I ) = I
76eqimssi 3027 . 2 ( I ∘ I ) ⊆ I
8 df-fun 4932 . 2 (Fun I ↔ (Rel I ∧ ( I ∘ I ) ⊆ I ))
91, 7, 8mpbir2an 860 1 Fun I
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wss 2945   I cid 4053  ccnv 4372  ccom 4377  Rel wrel 4378  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  cnvresid  5001  fnresi  5044  fvi  5258  ssdomg  6289  climshft2  10058
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