Theorem fnresi 6611

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 6610	. 2 ⊢ I Fn V
2		ssv 3960	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6605	. 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 3903   I cid 5513   ↾ cres 5621   Fn wfn 6477

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-fun 6484  df-fn 6485

This theorem is referenced by:  f1oi  6802  fninfp  7110  fndifnfp  7112  fnnfpeq0  7114  fveqf1o  7239  weniso  7291  iordsmo  8280  fipreima  9248  dfac9  10031  smndex1n0mnd  18786  pmtrfinv  19340  psdmplcl  22047  ustuqtop3  24129  fta1blem  26074  qaa  26229  dfiop2  31697  symgcom2  33026  tocycfvres1  33052  tocycfvres2  33053  cvmliftlem4  35261  cvmliftlem5  35262  poimirlem15  37615  poimirlem22  37622  ltrnid  40114  dvsid  44304  cjnpoly  46873  dflinc2  48395  tposideq  48872