Theorem fnresi 6627

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 6626	. 2 ⊢ I Fn V
2		ssv 3946	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6621	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 693	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3429   ⊆ wss 3889   I cid 5525   ↾ cres 5633   Fn wfn 6493

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-fun 6500  df-fn 6501

This theorem is referenced by:  f1oi  6818  f1oiOLD  6819  fninfp  7129  fndifnfp  7131  fnnfpeq0  7133  fveqf1o  7257  weniso  7309  iordsmo  8297  fipreima  9268  dfac9  10059  smndex1n0mnd  18883  pmtrfinv  19436  psdmplcl  22128  ustuqtop3  24208  fta1blem  26136  qaa  26289  dfiop2  31824  symgcom2  33145  tocycfvres1  33171  tocycfvres2  33172  cvmliftlem4  35470  cvmliftlem5  35471  poimirlem15  37956  poimirlem22  37963  ltrnid  40581  dvsid  44758  cjnpoly  47337  dflinc2  48886  tposideq  49363