Theorem fnresi 6668

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 6667	. 2 ⊢ I Fn V
2		ssv 3955	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6662	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 705	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

This proof depends on syntax axioms:  Vcvv 3451   ⊆ wss 3899   I cid 5545   ↾ cres 5653   Fn wfn 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-fun 6540  df-fn 6541

This theorem is used by:  f1oi  6863  f1oiOLD  6864  fninfp  7179  fndifnfp  7181  fnnfpeq0  7183  fveqf1o  7310  weniso  7364  iordsmo  8365  fipreima  9347  dfac9  10215  smndex1n0mnd  19111  pmtrfinv  19675  psdmplcl  22483  ustuqtop3  24562  fta1blem  26489  qaa  26647  dfiop2  32355  symgcom2  33645  tocycfvres1  33671  tocycfvres2  33672  cvmliftlem4  36053  cvmliftlem5  36054  poimirlem15  38553  poimirlem22  38560  ltrnid  41192  dvsid  45314  cjnpoly  47938  dflinc2  49521  tposideq  49995