MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resiexd Structured version   Visualization version   GIF version

Theorem resiexd 6632
Description: The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
Hypothesis
Ref Expression
resiexd.b (𝜑𝐵𝑉)
Assertion
Ref Expression
resiexd (𝜑 → ( I ↾ 𝐵) ∈ V)

Proof of Theorem resiexd
StepHypRef Expression
1 funi 6069 . 2 Fun I
2 resiexd.b . 2 (𝜑𝐵𝑉)
3 resfunexg 6631 . 2 ((Fun I ∧ 𝐵𝑉) → ( I ↾ 𝐵) ∈ V)
41, 2, 3sylancr 698 1 (𝜑 → ( I ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2127  Vcvv 3328   I cid 5161  cres 5256  Fun wfun 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045
This theorem is referenced by:  estrcid  16946  funcestrcsetclem4  16955  funcestrcsetclem5  16956  funcsetcestrclem4  16970  funcsetcestrclem5  16971  cusgrsize  26531  rclexi  38393  cnvrcl0  38403  dfrtrcl5  38407  relexp01min  38476  uspgrsprfo  42235  funcrngcsetc  42477  funcrngcsetcALT  42478  funcringcsetc  42514  funcringcsetcALTV2lem4  42518  funcringcsetcALTV2lem5  42519  funcringcsetclem4ALTV  42541  funcringcsetclem5ALTV  42542  rhmsubcALTVlem3  42585
  Copyright terms: Public domain W3C validator