Theorem fnresi 6622

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 6621	. 2 ⊢ I Fn V
2		ssv 3947	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6616	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 693	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3430   ⊆ wss 3890   I cid 5519   ↾ cres 5627   Fn wfn 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-fun 6495  df-fn 6496

This theorem is referenced by:  f1oi  6813  f1oiOLD  6814  fninfp  7123  fndifnfp  7125  fnnfpeq0  7127  fveqf1o  7251  weniso  7303  iordsmo  8291  fipreima  9262  dfac9  10053  smndex1n0mnd  18877  pmtrfinv  19430  psdmplcl  22141  ustuqtop3  24221  fta1blem  26149  qaa  26303  dfiop2  31842  symgcom2  33163  tocycfvres1  33189  tocycfvres2  33190  cvmliftlem4  35489  cvmliftlem5  35490  poimirlem15  37973  poimirlem22  37980  ltrnid  40598  dvsid  44779  cjnpoly  47352  dflinc2  48901  tposideq  49378