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Theorem f1ovi 6646
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 6645 . 2 ( I ↾ V):V–1-1-onto→V
2 reli 5691 . . . 4 Rel I
3 dfrel3 6048 . . . 4 (Rel I ↔ ( I ↾ V) = I )
42, 3mpbi 231 . . 3 ( I ↾ V) = I
5 f1oeq1 6597 . . 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 231 1 I :V–1-1-onto→V
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  Vcvv 3492   I cid 5452  cres 5550  Rel wrel 5553  1-1-ontowf1o 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355
This theorem is referenced by:  ncanth  7101
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