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Theorem f1ovi 6886
Description: The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
Assertion
Ref Expression
f1ovi I :V–1-1-onto→V

Proof of Theorem f1ovi
StepHypRef Expression
1 f1oi 6885 . 2 ( I ↾ V):V–1-1-onto→V
2 reli 5835 . . . 4 Rel I
3 dfrel3 6217 . . . 4 (Rel I ↔ ( I ↾ V) = I )
42, 3mpbi 230 . . 3 ( I ↾ V) = I
5 f1oeq1 6835 . . 3 (( I ↾ V) = I → (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V))
64, 5ax-mp 5 . 2 (( I ↾ V):V–1-1-onto→V ↔ I :V–1-1-onto→V)
71, 6mpbi 230 1 I :V–1-1-onto→V
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  Vcvv 3479   I cid 5576  cres 5686  Rel wrel 5689  1-1-ontowf1o 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567
This theorem is referenced by:  ncanth  7387
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