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 3960	. 2 ⊢ 𝐴 ⊆ V
3		fnssres 6644	. 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 702	1 ⊢ ( I ↾ 𝐴) Fn 𝐴

Syntax hints:  Vcvv 3454   ⊆ wss 3904   I cid 5541   ↾ cres 5649   Fn wfn 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-fun 6523  df-fn 6524

This theorem is referenced by:  f1oi  6845  f1oiOLD  6846  fninfp  7158  fndifnfp  7160  fnnfpeq0  7162  fveqf1o  7286  weniso  7338  iordsmo  8328  fipreima  9301  dfac9  10093  smndex1n0mnd  18949  pmtrfinv  19501  psdmplcl  22224  ustuqtop3  24300  fta1blem  26228  qaa  26384  dfiop2  31953  symgcom2  33261  tocycfvres1  33287  tocycfvres2  33288  cvmliftlem4  35635  cvmliftlem5  35636  poimirlem15  38131  poimirlem22  38138  ltrnid  40756  dvsid  44904  cjnpoly  47480  dflinc2  49029  tposideq  49506