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

Theorem fnresi 6466
 Description: The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.)
Assertion
Ref Expression
fnresi ( I ↾ 𝐴) Fn 𝐴

Proof of Theorem fnresi
StepHypRef Expression
1 idfn 6465 . 2 I Fn V
2 ssv 3978 . 2 𝐴 ⊆ V
3 fnssres 6460 . 2 (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
41, 2, 3mp2an 691 1 ( I ↾ 𝐴) Fn 𝐴
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3481   ⊆ wss 3920   I cid 5447   ↾ cres 5545   Fn wfn 6339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3483  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-res 5555  df-fun 6346  df-fn 6347 This theorem is referenced by:  f1oi  6644  fninfp  6928  fndifnfp  6930  fnnfpeq0  6932  fveqf1o  7052  weniso  7101  iordsmo  7991  fipreima  8828  dfac9  9561  smndex1n0mnd  18080  pmtrfinv  18592  ustuqtop3  22855  fta1blem  24775  qaa  24925  dfiop2  29542  symgcom2  30763  tocycfvres1  30787  tocycfvres2  30788  cvmliftlem4  32595  cvmliftlem5  32596  poimirlem15  35018  poimirlem22  35025  ltrnid  37377  dvsid  40956  dflinc2  44746
 Copyright terms: Public domain W3C validator