Theorem f1oi 5480

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 5315	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5272	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5292	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 145	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5450	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 937	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4273   `'ccnv 4610   |` cres 4613   Fn wfn 5193   -1-1-onto->wf1o 5197

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1ovi  5481  isoid  5789  enrefg  6742  ssdomg  6756  omp1eomlem  7071  ctm  7086  omct  7094  ctssexmid  7126  ssomct  12400  idmhm  12692  ssidcn  13004  dvid  13456  dvexp  13469  subctctexmid  14034