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Theorem resiexd 7218
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 6581 . 2 Fun I
2 resiexd.b . 2 (𝜑𝐵𝑉)
3 resfunexg 7217 . 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 3475   I cid 5574  cres 5679  Fun wfun 6538
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 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  setcid  18036  estrcid  18085  funcestrcsetclem5  18096  funcsetcestrclem5  18111  cusgrsize  28711  tocycfv  32268  tocycf  32276  lindspropd  32499  rclexi  42366  cnvrcl0  42376  dfrtrcl5  42380  relexp01min  42464  fundcmpsurbijinjpreimafv  46075  fundcmpsurinjALT  46080  isomgreqve  46493  ushrisomgr  46509  uspgrsprfo  46526  funcrngcsetc  46896  funcrngcsetcALT  46897  funcringcsetc  46933  funcringcsetcALTV2lem4  46937  funcringcsetcALTV2lem5  46938  funcringcsetclem4ALTV  46960  funcringcsetclem5ALTV  46961  rhmsubcALTVlem3  47004  itcoval0  47348  itcoval1  47349
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