Theorem fnresi 6689

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 6688	. 2 ⊢ I Fn V
2		ssv 4006	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6683	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 690	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3473   ⊆ wss 3949   I cid 5579   ↾ cres 5684   Fn wfn 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-fun 6555  df-fn 6556

This theorem is referenced by:  f1oi  6882  fninfp  7189  fndifnfp  7191  fnnfpeq0  7193  fveqf1o  7318  weniso  7368  iordsmo  8384  fipreima  9390  dfac9  10167  smndex1n0mnd  18871  pmtrfinv  19423  psdmplcl  22093  ustuqtop3  24168  fta1blem  26125  qaa  26278  dfiop2  31583  symgcom2  32828  tocycfvres1  32852  tocycfvres2  32853  cvmliftlem4  34931  cvmliftlem5  34932  poimirlem15  37141  poimirlem22  37148  ltrnid  39640  dvsid  43799  dflinc2  47556

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