Theorem fnresi 6673

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 6672	. 2 ⊢ I Fn V
2		ssv 3933	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6667	. 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 3450   ⊆ wss 3896   I cid 5545   ↾ cres 5653   Fn wfn 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-fun 6545  df-fn 6546

This theorem is used by:  f1oi  6868  f1oiOLD  6869  fninfp  7183  fndifnfp  7185  fnnfpeq0  7187  fveqf1o  7314  weniso  7368  iordsmo  8372  fipreima  9354  dfac9  10222  smndex1n0mnd  19120  pmtrfinv  19684  psdmplcl  22492  ustuqtop3  24571  fta1blem  26498  qaa  26656  dfiop2  32364  symgcom2  33695  tocycfvres1  33721  tocycfvres2  33722  cvmliftlem4  36143  cvmliftlem5  36144  poimirlem15  38662  poimirlem22  38669  ltrnid  41301  dvsid  45424  cjnpoly  48047  dflinc2  49630  tposideq  50109