Theorem fnresi 6610

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 6609	. 2 ⊢ I Fn V
2		ssv 3954	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6604	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3436   ⊆ wss 3897   I cid 5508   ↾ cres 5616   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-fun 6483  df-fn 6484

This theorem is referenced by:  f1oi  6801  fninfp  7108  fndifnfp  7110  fnnfpeq0  7112  fveqf1o  7236  weniso  7288  iordsmo  8277  fipreima  9242  dfac9  10028  smndex1n0mnd  18820  pmtrfinv  19373  psdmplcl  22077  ustuqtop3  24158  fta1blem  26103  qaa  26258  dfiop2  31733  symgcom2  33053  tocycfvres1  33079  tocycfvres2  33080  cvmliftlem4  35332  cvmliftlem5  35333  poimirlem15  37683  poimirlem22  37690  ltrnid  40182  dvsid  44372  cjnpoly  46928  dflinc2  48450  tposideq  48927