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Theorem fnresi 5698
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
StepHypRef Expression
1 funi 5618 . . 3  |-  Fun  _I
2 funres 5627 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 5 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5329 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5591 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 918 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    _I cid 4790   dom cdm 4999    |` cres 5001   Fun wfun 5582    Fn wfn 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-fun 5590  df-fn 5591
This theorem is referenced by:  f1oi  5851  fninfp  6088  fndifnfp  6090  fnnfpeq0  6092  fveqf1o  6193  weniso  6238  iordsmo  7028  fipreima  7826  dfac9  8516  pmtrfinv  16292  ustuqtop3  20509  fta1blem  22332  qaa  22481  dfiop2  26376  idssxp  27170  cvmliftlem4  28401  cvmliftlem5  28402  dvsid  30864  dflinc2  32110  ltrnid  34949
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