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Theorem resiexd 6710
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 6134 . 2 Fun I
2 resiexd.b . 2 (𝜑𝐵𝑉)
3 resfunexg 6709 . 2 ((Fun I ∧ 𝐵𝑉) → ( I ↾ 𝐵) ∈ V)
41, 2, 3sylancr 582 1 (𝜑 → ( I ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  Vcvv 3386   I cid 5220  cres 5315  Fun wfun 6096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110
This theorem is referenced by:  estrcid  17087  funcestrcsetclem4  17097  funcestrcsetclem5  17098  funcsetcestrclem4  17112  funcsetcestrclem5  17113  cusgrsize  26703  rclexi  38700  cnvrcl0  38710  dfrtrcl5  38714  relexp01min  38783  isomgreqve  42490  ushrisomgr  42506  uspgrsprfo  42550  funcrngcsetc  42792  funcrngcsetcALT  42793  funcringcsetc  42829  funcringcsetcALTV2lem4  42833  funcringcsetcALTV2lem5  42834  funcringcsetclem4ALTV  42856  funcringcsetclem5ALTV  42857  rhmsubcALTVlem3  42900
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