Theorem fnresi 6650

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 6649	. 2 ⊢ I Fn V
2		ssv 3974	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6644	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3450   ⊆ wss 3917   I cid 5535   ↾ cres 5643   Fn wfn 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-fun 6516  df-fn 6517

This theorem is referenced by:  f1oi  6841  fninfp  7151  fndifnfp  7153  fnnfpeq0  7155  fveqf1o  7280  weniso  7332  iordsmo  8329  fipreima  9316  dfac9  10097  smndex1n0mnd  18846  pmtrfinv  19398  psdmplcl  22056  ustuqtop3  24138  fta1blem  26083  qaa  26238  dfiop2  31689  symgcom2  33048  tocycfvres1  33074  tocycfvres2  33075  cvmliftlem4  35282  cvmliftlem5  35283  poimirlem15  37636  poimirlem22  37643  ltrnid  40136  dvsid  44327  dflinc2  48403  tposideq  48880