Theorem fnresi 6680

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 6679	. 2 ⊢ I Fn V
2		ssv 4007	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6674	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 691	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3475   ⊆ wss 3949   I cid 5574   ↾ cres 5679   Fn wfn 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-fun 6546  df-fn 6547

This theorem is referenced by:  f1oi  6872  fninfp  7172  fndifnfp  7174  fnnfpeq0  7176  fveqf1o  7301  weniso  7351  iordsmo  8357  fipreima  9358  dfac9  10131  smndex1n0mnd  18793  pmtrfinv  19329  ustuqtop3  23748  fta1blem  25686  qaa  25836  dfiop2  31006  symgcom2  32245  tocycfvres1  32269  tocycfvres2  32270  cvmliftlem4  34279  cvmliftlem5  34280  poimirlem15  36503  poimirlem22  36510  ltrnid  39006  dvsid  43090  dflinc2  47091