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Theorem fnresi 6631
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 6630 . 2 I Fn V
2 ssv 3969 . 2 𝐴 ⊆ V
3 fnssres 6625 . 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 3444  wss 3911   I cid 5531  cres 5636   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-fun 6499  df-fn 6500
This theorem is referenced by:  f1oi  6823  fninfp  7121  fndifnfp  7123  fnnfpeq0  7125  fveqf1o  7250  weniso  7300  iordsmo  8304  fipreima  9305  dfac9  10077  smndex1n0mnd  18727  pmtrfinv  19248  ustuqtop3  23611  fta1blem  25549  qaa  25699  dfiop2  30737  symgcom2  31984  tocycfvres1  32008  tocycfvres2  32009  cvmliftlem4  33939  cvmliftlem5  33940  poimirlem15  36139  poimirlem22  36146  ltrnid  38644  dvsid  42699  dflinc2  46577
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