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Theorem resiexd 7220
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 6580 . 2 Fun I
2 resiexd.b . 2 (𝜑𝐵𝑉)
3 resfunexg 7219 . 2 ((Fun I ∧ 𝐵𝑉) → ( I ↾ 𝐵) ∈ V)
41, 2, 3sylancr 587 1 (𝜑 → ( I ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3474   I cid 5573  cres 5678  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  setcid  18038  estrcid  18087  funcestrcsetclem5  18098  funcsetcestrclem5  18113  cusgrsize  28749  tocycfv  32309  tocycf  32317  lindspropd  32544  rclexi  42448  cnvrcl0  42458  dfrtrcl5  42462  relexp01min  42546  fundcmpsurbijinjpreimafv  46154  fundcmpsurinjALT  46159  isomgreqve  46572  ushrisomgr  46588  uspgrsprfo  46605  funcrngcsetc  46975  funcrngcsetcALT  46976  funcringcsetc  47012  funcringcsetcALTV2lem4  47016  funcringcsetcALTV2lem5  47017  funcringcsetclem4ALTV  47039  funcringcsetclem5ALTV  47040  rhmsubcALTVlem3  47083  itcoval0  47426  itcoval1  47427
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