Theorem fnresi 6664

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 6663	. 2 ⊢ I Fn V
2		ssv 3961	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6658	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 704	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3455   ⊆ wss 3905   I cid 5555   ↾ cres 5663   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-fun 6538  df-fn 6539

This theorem is referenced by:  f1oi  6859  f1oiOLD  6860  fninfp  7172  fndifnfp  7174  fnnfpeq0  7176  fveqf1o  7300  weniso  7352  iordsmo  8340  fipreima  9311  dfac9  10116  smndex1n0mnd  18969  pmtrfinv  19526  psdmplcl  22325  ustuqtop3  24400  fta1blem  26328  qaa  26484  dfiop2  32105  symgcom2  33404  tocycfvres1  33430  tocycfvres2  33431  cvmliftlem4  35780  cvmliftlem5  35781  poimirlem15  38286  poimirlem22  38293  ltrnid  40909  dvsid  45041  cjnpoly  47626  dflinc2  49190  tposideq  49666