Theorem fnresi 6667

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 6666	. 2 ⊢ I Fn V
2		ssv 3983	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6661	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3459   ⊆ wss 3926   I cid 5547   ↾ cres 5656   Fn wfn 6526

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-fun 6533  df-fn 6534

This theorem is referenced by:  f1oi  6856  fninfp  7166  fndifnfp  7168  fnnfpeq0  7170  fveqf1o  7295  weniso  7347  iordsmo  8371  fipreima  9370  dfac9  10151  smndex1n0mnd  18890  pmtrfinv  19442  psdmplcl  22100  ustuqtop3  24182  fta1blem  26128  qaa  26283  dfiop2  31734  symgcom2  33095  tocycfvres1  33121  tocycfvres2  33122  cvmliftlem4  35310  cvmliftlem5  35311  poimirlem15  37659  poimirlem22  37666  ltrnid  40154  dvsid  44355  dflinc2  48386  tposideq  48863