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Theorem fnresi 6169
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 6081 . . 3 Fun I
2 funres 6090 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 5615 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 6052 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 993 1 ( I ↾ 𝐴) Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632   I cid 5173  dom cdm 5266  cres 5268  Fun wfun 6043   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-fun 6051  df-fn 6052
This theorem is referenced by:  idssxp  6170  f1oi  6336  fninfp  6605  fndifnfp  6607  fnnfpeq0  6609  fveqf1o  6721  weniso  6768  iordsmo  7624  fipreima  8439  dfac9  9170  pmtrfinv  18101  ustuqtop3  22268  fta1blem  24147  qaa  24297  dfiop2  28942  cvmliftlem4  31598  cvmliftlem5  31599  poimirlem15  33755  poimirlem22  33762  ltrnid  35942  rtrclex  38444  dvsid  39050  dflinc2  42727
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