MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnresi Structured version   Visualization version   GIF version

Theorem fnresi 5907
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 5819 . . 3 Fun I
2 funres 5828 . . 3 (Fun I → Fun ( I ↾ 𝐴))
31, 2ax-mp 5 . 2 Fun ( I ↾ 𝐴)
4 dmresi 5362 . 2 dom ( I ↾ 𝐴) = 𝐴
5 df-fn 5792 . 2 (( I ↾ 𝐴) Fn 𝐴 ↔ (Fun ( I ↾ 𝐴) ∧ dom ( I ↾ 𝐴) = 𝐴))
63, 4, 5mpbir2an 956 1 ( I ↾ 𝐴) Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474   I cid 4937  dom cdm 5027  cres 5029  Fun wfun 5783   Fn wfn 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-res 5039  df-fun 5791  df-fn 5792
This theorem is referenced by:  f1oi  6070  fninfp  6322  fndifnfp  6324  fnnfpeq0  6326  fveqf1o  6434  weniso  6481  iordsmo  7318  fipreima  8132  dfac9  8818  pmtrfinv  17652  ustuqtop3  21804  fta1blem  23676  qaa  23826  dfiop2  27789  idssxp  28604  cvmliftlem4  30317  cvmliftlem5  30318  poimirlem15  32377  poimirlem22  32384  ltrnid  34222  rtrclex  36726  dvsid  37335  dflinc2  41974
  Copyright terms: Public domain W3C validator