Theorem fnresi 6545

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 6544	. 2 ⊢ I Fn V
2		ssv 3941	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6539	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 688	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3422   ⊆ wss 3883   I cid 5479   ↾ cres 5582   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-fun 6420  df-fn 6421

This theorem is referenced by:  f1oi  6737  fninfp  7028  fndifnfp  7030  fnnfpeq0  7032  fveqf1o  7155  weniso  7205  iordsmo  8159  fipreima  9055  dfac9  9823  smndex1n0mnd  18466  pmtrfinv  18984  ustuqtop3  23303  fta1blem  25238  qaa  25388  dfiop2  30016  symgcom2  31255  tocycfvres1  31279  tocycfvres2  31280  cvmliftlem4  33150  cvmliftlem5  33151  poimirlem15  35719  poimirlem22  35726  ltrnid  38076  dvsid  41838  dflinc2  45639