Theorem fnresi 6647

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 6646	. 2 ⊢ I Fn V
2		ssv 3971	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6641	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3447   ⊆ wss 3914   I cid 5532   ↾ cres 5640   Fn wfn 6506

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-fun 6513  df-fn 6514

This theorem is referenced by:  f1oi  6838  fninfp  7148  fndifnfp  7150  fnnfpeq0  7152  fveqf1o  7277  weniso  7329  iordsmo  8326  fipreima  9309  dfac9  10090  smndex1n0mnd  18839  pmtrfinv  19391  psdmplcl  22049  ustuqtop3  24131  fta1blem  26076  qaa  26231  dfiop2  31682  symgcom2  33041  tocycfvres1  33067  tocycfvres2  33068  cvmliftlem4  35275  cvmliftlem5  35276  poimirlem15  37629  poimirlem22  37636  ltrnid  40129  dvsid  44320  cjnpoly  46890  dflinc2  48399  tposideq  48876