Theorem fnresi 6580

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 6579	. 2 ⊢ I Fn V
2		ssv 3947	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6574	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 688	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3434   ⊆ wss 3889   I cid 5490   ↾ cres 5593   Fn wfn 6442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-br 5078  df-opab 5140  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-res 5603  df-fun 6449  df-fn 6450

This theorem is referenced by:  f1oi  6772  fninfp  7066  fndifnfp  7068  fnnfpeq0  7070  fveqf1o  7195  weniso  7245  iordsmo  8208  fipreima  9153  dfac9  9920  smndex1n0mnd  18579  pmtrfinv  19097  ustuqtop3  23423  fta1blem  25361  qaa  25511  dfiop2  30143  symgcom2  31381  tocycfvres1  31405  tocycfvres2  31406  cvmliftlem4  33278  cvmliftlem5  33279  poimirlem15  35820  poimirlem22  35827  ltrnid  38175  dvsid  41973  dflinc2  45791