Theorem f1oi 5677

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 5499	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5453	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5473	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 146	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5645	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 955	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4431   `'ccnv 4771   |` cres 4774   Fn wfn 5370   -1-1-onto->wf1o 5374

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382

This theorem is referenced by:  f1ovi  5678  isoid  6009  enrefg  7043  ssdomg  7058  omp1eomlem  7427  ctm  7442  omct  7450  ctssexmid  7483  ssomct  13317  idmhm  13756  idghm  14042  gzsumgsum1  14133  ssidcn  15237  dvid  15722  dvidre  15724  dvexp  15738  ausgrusgrben  16326  subctctexmid  16947