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Theorem f1ovi 6482
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 6481 . 2 ( I ↾ V):V–1-1-onto→V
2 reli 5548 . . . 4 Rel I
3 dfrel3 5894 . . . 4 (Rel I ↔ ( I ↾ V) = I )
42, 3mpbi 222 . . 3 ( I ↾ V) = I
5 f1oeq1 6433 . . 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 222 1 I :V–1-1-onto→V
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1507  Vcvv 3415   I cid 5311  cres 5409  Rel wrel 5412  1-1-ontowf1o 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195
This theorem is referenced by:  ncanth  6935
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