Theorem fnresi 6709

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 6708	. 2 ⊢ I Fn V
2		ssv 4033	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6703	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 691	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3488   ⊆ wss 3976   I cid 5592   ↾ cres 5702   Fn wfn 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-fun 6575  df-fn 6576

This theorem is referenced by:  f1oi  6900  fninfp  7208  fndifnfp  7210  fnnfpeq0  7212  fveqf1o  7338  weniso  7390  iordsmo  8413  fipreima  9428  dfac9  10206  smndex1n0mnd  18947  pmtrfinv  19503  psdmplcl  22189  ustuqtop3  24273  fta1blem  26230  qaa  26383  dfiop2  31785  symgcom2  33077  tocycfvres1  33103  tocycfvres2  33104  cvmliftlem4  35256  cvmliftlem5  35257  poimirlem15  37595  poimirlem22  37602  ltrnid  40092  dvsid  44300  dflinc2  48139