Theorem fnresi 6697

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 6696	. 2 ⊢ I Fn V
2		ssv 4008	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6691	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3480   ⊆ wss 3951   I cid 5577   ↾ cres 5687   Fn wfn 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-fn 6564

This theorem is referenced by:  f1oi  6886  fninfp  7194  fndifnfp  7196  fnnfpeq0  7198  fveqf1o  7322  weniso  7374  iordsmo  8397  fipreima  9398  dfac9  10177  smndex1n0mnd  18925  pmtrfinv  19479  psdmplcl  22166  ustuqtop3  24252  fta1blem  26210  qaa  26365  dfiop2  31772  symgcom2  33104  tocycfvres1  33130  tocycfvres2  33131  cvmliftlem4  35293  cvmliftlem5  35294  poimirlem15  37642  poimirlem22  37649  ltrnid  40137  dvsid  44350  dflinc2  48327  tposideq  48788