Theorem fnresi 6619

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 6618	. 2 ⊢ I Fn V
2		ssv 3956	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6613	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3438   ⊆ wss 3899   I cid 5516   ↾ cres 5624   Fn wfn 6485

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-fun 6492  df-fn 6493

This theorem is referenced by:  f1oi  6810  f1oiOLD  6811  fninfp  7118  fndifnfp  7120  fnnfpeq0  7122  fveqf1o  7246  weniso  7298  iordsmo  8287  fipreima  9256  dfac9  10045  smndex1n0mnd  18835  pmtrfinv  19388  psdmplcl  22103  ustuqtop3  24185  fta1blem  26130  qaa  26285  dfiop2  31777  symgcom2  33115  tocycfvres1  33141  tocycfvres2  33142  cvmliftlem4  35431  cvmliftlem5  35432  poimirlem15  37775  poimirlem22  37782  ltrnid  40334  dvsid  44514  cjnpoly  47077  dflinc2  48598  tposideq  49075