Theorem f1oi 5560

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 5393	. 2 |- ( _I |` A ) Fn A
2		cnvresid 5348	. . . 4 |- `' ( _I |` A ) = ( _I |` A )
3	2	fneq1i 5368	. . 3 |- ( `' ( _I |` A ) Fn A <-> ( _I |` A ) Fn A )
4	1, 3	mpbir 146	. 2 |- `' ( _I |` A ) Fn A
5		dff1o4 5530	. 2 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ `' ( _I |` A ) Fn A ) )
6	1, 4, 5	mpbir2an 945	1 |- ( _I |` A ) : A -1-1-onto-> A

Syntax hints:   _I cid 4335   `'ccnv 4674   |` cres 4677   Fn wfn 5266   -1-1-onto->wf1o 5270

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278

This theorem is referenced by:  f1ovi  5561  isoid  5879  enrefg  6855  ssdomg  6870  omp1eomlem  7196  ctm  7211  omct  7219  ctssexmid  7252  ssomct  12816  idmhm  13301  idghm  13595  ssidcn  14682  dvid  15167  dvidre  15169  dvexp  15183  subctctexmid  15937