Theorem fnresi 6698

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 6697	. 2 ⊢ I Fn V
2		ssv 3937	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6692	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 705	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3900   I cid 5552   ↾ cres 5660   Fn wfn 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-res 5670  df-fun 6568  df-fn 6569

This theorem is used by:  f1oi  6894  f1oiOLD  6895  fninfp  7210  fndifnfp  7212  fnnfpeq0  7214  fveqf1o  7341  weniso  7395  iordsmo  8403  fipreima  9385  dfac9  10254  smndex1n0mnd  19161  pmtrfinv  19726  psdmplcl  22538  ustuqtop3  24617  fta1blem  26544  qaa  26704  dfiop2  32412  symgcom2  33744  tocycfvres1  33770  tocycfvres2  33771  cvmliftlem4  36261  cvmliftlem5  36262  poimirlem15  38821  poimirlem22  38828  ltrnid  41458  dvsid  45576  cjnpoly  48212  dflinc2  49853  tposideq  50332