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Theorem fnresi 6167
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 6079 . . 3 Fun I
2 funres 6088 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 5613 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 6050 . 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 1630   I cid 5171  dom cdm 5264  cres 5266  Fun wfun 6041   Fn wfn 6042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-res 5276  df-fun 6049  df-fn 6050
This theorem is referenced by:  idssxp  6168  f1oi  6333  fninfp  6602  fndifnfp  6604  fnnfpeq0  6606  fveqf1o  6718  weniso  6765  iordsmo  7621  fipreima  8435  dfac9  9148  pmtrfinv  18079  ustuqtop3  22246  fta1blem  24125  qaa  24275  dfiop2  28919  cvmliftlem4  31575  cvmliftlem5  31576  poimirlem15  33735  poimirlem22  33742  ltrnid  35922  rtrclex  38424  dvsid  39030  dflinc2  42707
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