Theorem f1oi 5469

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 5304	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5261	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5281	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 145	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5439	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 932	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4265   `'ccnv 4602   |` cres 4605   Fn wfn 5182   -1-1-onto->wf1o 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194

This theorem is referenced by:  f1ovi  5470  isoid  5777  enrefg  6726  ssdomg  6740  omp1eomlem  7055  ctm  7070  omct  7078  ctssexmid  7110  ssomct  12374  ssidcn  12810  dvid  13262  dvexp  13275  subctctexmid  13841