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Theorem restid 16765
 Description: The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypothesis
Ref Expression
restid.1 𝑋 = 𝐽
Assertion
Ref Expression
restid (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Proof of Theorem restid
StepHypRef Expression
1 restid.1 . . 3 𝑋 = 𝐽
2 uniexg 7464 . . 3 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2856 . 2 (𝐽𝑉𝑋 ∈ V)
41eqimss2i 3951 . . 3 𝐽𝑋
5 sspwuni 4987 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
64, 5mpbir 234 . 2 𝐽 ⊆ 𝒫 𝑋
7 restid2 16762 . 2 ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽t 𝑋) = 𝐽)
83, 6, 7sylancl 589 1 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798  (class class class)co 7150   ↾t crest 16752 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rest 16754 This theorem is referenced by:  toponrestid  21621  restin  21866  cnrmnrm  22061  cmpkgen  22251  xkopt  22355  xkoinjcn  22387  ussid  22961  tuslem  22968  cnperf  23521  retopconn  23530  abscncfALT  23625  cnmpopc  23629  recnperf  24604  lhop1lem  24712  cxpcn3  25436  retopsconn  32727  ivthALT  34073  binomcxplemdvbinom  41430  binomcxplemnotnn0  41433  fsumcncf  42886  ioccncflimc  42893  cncfuni  42894  icocncflimc  42897  cncfiooicclem1  42901  itgsubsticclem  42983  dirkercncflem2  43112  dirkercncflem4  43114  fourierdlem32  43147  fourierdlem33  43148  fourierdlem62  43176  fourierdlem93  43207  fourierdlem101  43215
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