Theorem fnresi 6698

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

Step	Hyp	Ref	Expression
1		idfn 6697	. 2 ⊢ I Fn V
2		ssv 4020	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6692	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3478   ⊆ wss 3963   I cid 5582   ↾ cres 5691   Fn wfn 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565  df-fn 6566

This theorem is referenced by:  f1oi  6887  fninfp  7194  fndifnfp  7196  fnnfpeq0  7198  fveqf1o  7322  weniso  7374  iordsmo  8396  fipreima  9396  dfac9  10175  smndex1n0mnd  18938  pmtrfinv  19494  psdmplcl  22184  ustuqtop3  24268  fta1blem  26225  qaa  26380  dfiop2  31782  symgcom2  33087  tocycfvres1  33113  tocycfvres2  33114  cvmliftlem4  35273  cvmliftlem5  35274  poimirlem15  37622  poimirlem22  37629  ltrnid  40118  dvsid  44327  dflinc2  48256