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Theorem f1oi 5291
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 5131 . 2  |-  (  _I  |`  A )  Fn  A
2 cnvresid 5088 . . . 4  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
32fneq1i 5108 . . 3  |-  ( `' (  _I  |`  A )  Fn  A  <->  (  _I  |`  A )  Fn  A
)
41, 3mpbir 144 . 2  |-  `' (  _I  |`  A )  Fn  A
5 dff1o4 5261 . 2  |-  ( (  _I  |`  A ) : A -1-1-onto-> A  <->  ( (  _I  |`  A )  Fn  A  /\  `' (  _I  |`  A )  Fn  A ) )
61, 4, 5mpbir2an 888 1  |-  (  _I  |`  A ) : A -1-1-onto-> A
Colors of variables: wff set class
Syntax hints:    _I cid 4115   `'ccnv 4437    |` cres 4440    Fn wfn 5010   -1-1-onto->wf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022
This theorem is referenced by:  f1ovi  5292  isoid  5589  enrefg  6481  ssdomg  6495
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