Theorem f1oi 5501

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

Step	Hyp	Ref	Expression
1		fnresi 5335	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5292	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5312	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 146	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5471	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 942	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4290   `'ccnv 4627   |` cres 4630   Fn wfn 5213   -1-1-onto->wf1o 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225

This theorem is referenced by:  f1ovi  5502  isoid  5813  enrefg  6766  ssdomg  6780  omp1eomlem  7095  ctm  7110  omct  7118  ctssexmid  7150  ssomct  12448  idmhm  12865  ssidcn  13749  dvid  14201  dvexp  14214  subctctexmid  14789