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Theorem resiexd 7167
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 6534 . 2 Fun I
2 resiexd.b . 2 (𝜑𝐵𝑉)
3 resfunexg 7166 . 2 ((Fun I ∧ 𝐵𝑉) → ( I ↾ 𝐵) ∈ V)
41, 2, 3sylancr 588 1 (𝜑 → ( I ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444   I cid 5531  cres 5636  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  setcid  17977  estrcid  18026  funcestrcsetclem5  18037  funcsetcestrclem5  18052  cusgrsize  28444  tocycfv  32007  tocycf  32015  lindspropd  32218  rclexi  41975  cnvrcl0  41985  dfrtrcl5  41989  relexp01min  42073  fundcmpsurbijinjpreimafv  45685  fundcmpsurinjALT  45690  isomgreqve  46103  ushrisomgr  46119  uspgrsprfo  46136  funcrngcsetc  46382  funcrngcsetcALT  46383  funcringcsetc  46419  funcringcsetcALTV2lem4  46423  funcringcsetcALTV2lem5  46424  funcringcsetclem4ALTV  46446  funcringcsetclem5ALTV  46447  rhmsubcALTVlem3  46490  itcoval0  46834  itcoval1  46835
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