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Theorem f1oi 5304
Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
f1oi ( I ↾ 𝐴):𝐴1-1-onto𝐴

Proof of Theorem f1oi
StepHypRef Expression
1 fnresi 5144 . 2 ( I ↾ 𝐴) Fn 𝐴
2 cnvresid 5101 . . . 4 ( I ↾ 𝐴) = ( I ↾ 𝐴)
32fneq1i 5121 . . 3 (( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴)
41, 3mpbir 145 . 2 ( I ↾ 𝐴) Fn 𝐴
5 dff1o4 5274 . 2 (( I ↾ 𝐴):𝐴1-1-onto𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴( I ↾ 𝐴) Fn 𝐴))
61, 4, 5mpbir2an 889 1 ( I ↾ 𝐴):𝐴1-1-onto𝐴
Colors of variables: wff set class
Syntax hints:   I cid 4124  ccnv 4451  cres 4454   Fn wfn 5023  1-1-ontowf1o 5027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035
This theorem is referenced by:  f1ovi  5305  isoid  5603  enrefg  6535  ssdomg  6549  ctm  6845
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