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Theorem fnresi 5922
 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 5834 . . 3 Fun I
2 funres 5843 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 5376 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 5807 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 957 1 ( I ↾ 𝐴) Fn 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   I cid 4948  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806  df-fn 5807 This theorem is referenced by:  f1oi  6086  fninfp  6345  fndifnfp  6347  fnnfpeq0  6349  fveqf1o  6457  weniso  6504  iordsmo  7341  fipreima  8155  dfac9  8841  pmtrfinv  17704  ustuqtop3  21857  fta1blem  23732  qaa  23882  dfiop2  27996  idssxp  28811  cvmliftlem4  30524  cvmliftlem5  30525  poimirlem15  32594  poimirlem22  32601  ltrnid  34439  rtrclex  36943  dvsid  37552  dflinc2  41993
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