Theorem fnresi 6665

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 6664	. 2 ⊢ I Fn V
2		ssv 3969	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6659	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 704	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3463   ⊆ wss 3913   I cid 5556   ↾ cres 5664   Fn wfn 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-fun 6539  df-fn 6540

This theorem is referenced by:  f1oi  6860  f1oiOLD  6861  fninfp  7173  fndifnfp  7175  fnnfpeq0  7177  fveqf1o  7301  weniso  7353  iordsmo  8343  fipreima  9314  dfac9  10119  smndex1n0mnd  18973  pmtrfinv  19530  psdmplcl  22293  ustuqtop3  24368  fta1blem  26296  qaa  26452  dfiop2  32045  symgcom2  33344  tocycfvres1  33370  tocycfvres2  33371  cvmliftlem4  35678  cvmliftlem5  35679  poimirlem15  38173  poimirlem22  38180  ltrnid  40798  dvsid  44932  cjnpoly  47514  dflinc2  49074  tposideq  49550