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Theorem f1ovi 6324
 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 6323 . 2 ( I ↾ V):V–1-1-onto→V
2 reli 5393 . . . 4 Rel I
3 dfrel3 5738 . . . 4 (Rel I ↔ ( I ↾ V) = I )
42, 3mpbi 220 . . 3 ( I ↾ V) = I
5 f1oeq1 6276 . . 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 220 1 I :V–1-1-onto→V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1620  Vcvv 3328   I cid 5161   ↾ cres 5256  Rel wrel 5259  –1-1-onto→wf1o 6036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044 This theorem is referenced by:  ncanth  6760
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