MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restid Structured version   Visualization version   GIF version

Theorem restid 16695
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 7456 . . 3 (𝐽𝑉 𝐽 ∈ V)
31, 2eqeltrid 2914 . 2 (𝐽𝑉𝑋 ∈ V)
41eqimss2i 4023 . . 3 𝐽𝑋
5 sspwuni 5013 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
64, 5mpbir 232 . 2 𝐽 ⊆ 𝒫 𝑋
7 restid2 16692 . 2 ((𝑋 ∈ V ∧ 𝐽 ⊆ 𝒫 𝑋) → (𝐽t 𝑋) = 𝐽)
83, 6, 7sylancl 586 1 (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684
This theorem is referenced by:  toponrestid  21457  restin  21702  cnrmnrm  21897  cmpkgen  22087  xkopt  22191  xkoinjcn  22223  ussid  22796  tuslem  22803  cnperf  23355  retopconn  23364  abscncfALT  23455  cnmpopc  23459  recnperf  24430  lhop1lem  24537  cxpcn3  25256  retopsconn  32393  ivthALT  33580  binomcxplemdvbinom  40562  binomcxplemnotnn0  40565  fsumcncf  42037  ioccncflimc  42044  cncfuni  42045  icocncflimc  42048  cncfiooicclem1  42052  itgsubsticclem  42136  dirkercncflem2  42266  dirkercncflem4  42268  fourierdlem32  42301  fourierdlem33  42302  fourierdlem62  42330  fourierdlem93  42361  fourierdlem101  42369
  Copyright terms: Public domain W3C validator