Theorem fnresi 6448

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 6447	. 2 ⊢ I Fn V
2		ssv 3939	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6442	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 691	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3441   ⊆ wss 3881   I cid 5424   ↾ cres 5521   Fn wfn 6319

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-fun 6326  df-fn 6327

This theorem is referenced by:  f1oi  6627  fninfp  6913  fndifnfp  6915  fnnfpeq0  6917  fveqf1o  7037  weniso  7086  iordsmo  7977  fipreima  8814  dfac9  9547  smndex1n0mnd  18069  pmtrfinv  18581  ustuqtop3  22849  fta1blem  24769  qaa  24919  dfiop2  29536  symgcom2  30778  tocycfvres1  30802  tocycfvres2  30803  cvmliftlem4  32648  cvmliftlem5  32649  poimirlem15  35072  poimirlem22  35079  ltrnid  37431  dvsid  41035  dflinc2  44819