Theorem fnresi 6666

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 6665	. 2 ⊢ I Fn V
2		ssv 3955	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6660	. 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 6532

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 6539  df-fn 6540

This theorem is used by:  f1oi  6861  f1oiOLD  6862  fninfp  7177  fndifnfp  7179  fnnfpeq0  7181  fveqf1o  7308  weniso  7362  iordsmo  8358  fipreima  9340  dfac9  10208  smndex1n0mnd  19104  pmtrfinv  19668  psdmplcl  22476  ustuqtop3  24555  fta1blem  26482  qaa  26640  dfiop2  32348  symgcom2  33638  tocycfvres1  33664  tocycfvres2  33665  cvmliftlem4  36032  cvmliftlem5  36033  poimirlem15  38533  poimirlem22  38540  ltrnid  41172  dvsid  45300  cjnpoly  47908  dflinc2  49491  tposideq  49965