Theorem f1oi 5545

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 5378	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5333	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5353	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 146	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5515	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 944	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4324   `'ccnv 4663   |` cres 4666   Fn wfn 5254   -1-1-onto->wf1o 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266

This theorem is referenced by:  f1ovi  5546  isoid  5860  enrefg  6832  ssdomg  6846  omp1eomlem  7169  ctm  7184  omct  7192  ctssexmid  7225  ssomct  12687  idmhm  13171  idghm  13465  ssidcn  14530  dvid  15015  dvidre  15017  dvexp  15031  subctctexmid  15731