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Theorem f1oi 5164
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 ) : A -1-1-onto-> A

Proof of Theorem f1oi
StepHypRef Expression
1 fnresi 5016 . 2  |-  (  _I  |`  A )  Fn  A
2 cnvresid 4973 . . . 4  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
32fneq1i 4993 . . 3  |-  ( `' (  _I  |`  A )  Fn  A  <->  (  _I  |`  A )  Fn  A
)
41, 3mpbir 134 . 2  |-  `' (  _I  |`  A )  Fn  A
5 dff1o4 5134 . 2  |-  ( (  _I  |`  A ) : A -1-1-onto-> A  <->  ( (  _I  |`  A )  Fn  A  /\  `' (  _I  |`  A )  Fn  A ) )
61, 4, 5mpbir2an 849 1  |-  (  _I  |`  A ) : A -1-1-onto-> A
Colors of variables: wff set class
Syntax hints:    _I cid 4025   `'ccnv 4344    |` cres 4347    Fn wfn 4897   -1-1-onto->wf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  f1ovi  5165  isoid  5450  enrefg  6244  ssdomg  6258
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