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Theorem f1ovi 6888
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 6887 . 2 ( I ↾ V):V–1-1-onto→V
2 reli 5839 . . . 4 Rel I
3 dfrel3 6220 . . . 4 (Rel I ↔ ( I ↾ V) = I )
42, 3mpbi 230 . . 3 ( I ↾ V) = I
5 f1oeq1 6837 . . 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 1537  Vcvv 3478   I cid 5582  cres 5691  Rel wrel 5694  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  ncanth  7386
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