Theorem fnresi 5441

Description: Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
fnresi	|- ( _I |` A ) Fn A

Proof of Theorem fnresi

Step	Hyp	Ref	Expression
1		funi 5350	. . 3 |- Fun _I
2		funres 5359	. . 3 |- ( Fun _I -> Fun ( _I |` A ) )
3	1, 2	ax-mp 5	. 2 |- Fun ( _I |` A )
4		dmresi 5060	. 2 |- dom ( _I |` A ) = A
5		df-fn 5321	. 2 |- ( ( _I |` A ) Fn A <-> ( Fun ( _I |` A ) /\ dom ( _I |` A ) = A ) )
6	3, 4, 5	mpbir2an 948	1 |- ( _I |` A ) Fn A

Syntax hints:   = wceq 1395   _I cid 4379   dom cdm 4719   |` cres 4721   Fun wfun 5312   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-res 4731  df-fun 5320  df-fn 5321

This theorem is referenced by:  f1oi  5611  iordsmo  6443  omp1eomlem  7261  ctm  7276  xnn0nnen  10659