Theorem fnresi 6678

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 6677	. 2 ⊢ I Fn V
2		ssv 4005	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6672	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 688	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3472   ⊆ wss 3947   I cid 5572   ↾ cres 5677   Fn wfn 6537

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-fun 6544  df-fn 6545

This theorem is referenced by:  f1oi  6870  fninfp  7173  fndifnfp  7175  fnnfpeq0  7177  fveqf1o  7303  weniso  7353  iordsmo  8359  fipreima  9360  dfac9  10133  smndex1n0mnd  18829  pmtrfinv  19370  ustuqtop3  23968  fta1blem  25921  qaa  26072  dfiop2  31273  symgcom2  32515  tocycfvres1  32539  tocycfvres2  32540  cvmliftlem4  34577  cvmliftlem5  34578  poimirlem15  36806  poimirlem22  36813  ltrnid  39309  dvsid  43392  dflinc2  47178