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Theorem fnresi 5711
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 5631 . . 3  |-  Fun  _I
2 funres 5640 . . 3  |-  ( Fun 
_I  ->  Fun  (  _I  |`  A ) )
31, 2ax-mp 5 . 2  |-  Fun  (  _I  |`  A )
4 dmresi 5180 . 2  |-  dom  (  _I  |`  A )  =  A
5 df-fn 5604 . 2  |-  ( (  _I  |`  A )  Fn  A  <->  ( Fun  (  _I  |`  A )  /\  dom  (  _I  |`  A )  =  A ) )
63, 4, 5mpbir2an 928 1  |-  (  _I  |`  A )  Fn  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    _I cid 4764   dom cdm 4854    |` cres 4856   Fun wfun 5595    Fn wfn 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-fun 5603  df-fn 5604
This theorem is referenced by:  f1oi  5866  fninfp  6106  fndifnfp  6108  fnnfpeq0  6110  fveqf1o  6215  weniso  6260  iordsmo  7084  fipreima  7886  dfac9  8564  pmtrfinv  17053  ustuqtop3  21189  fta1blem  22994  qaa  23144  dfiop2  27241  idssxp  28067  cvmliftlem4  29799  cvmliftlem5  29800  poimirlem15  31662  poimirlem22  31669  ltrnid  33412  dvsid  36320  dflinc2  38975
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