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Theorem elrestr 16692
Description: Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
elrestr ((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))

Proof of Theorem elrestr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . . 4 (𝐴𝑆) = (𝐴𝑆)
2 ineq1 4180 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑆) = (𝐴𝑆))
32rspceeqv 3637 . . . 4 ((𝐴𝐽 ∧ (𝐴𝑆) = (𝐴𝑆)) → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
41, 3mpan2 687 . . 3 (𝐴𝐽 → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
5 elrest 16691 . . 3 ((𝐽𝑉𝑆𝑊) → ((𝐴𝑆) ∈ (𝐽t 𝑆) ↔ ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆)))
64, 5syl5ibr 247 . 2 ((𝐽𝑉𝑆𝑊) → (𝐴𝐽 → (𝐴𝑆) ∈ (𝐽t 𝑆)))
763impia 1109 1 ((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3139  cin 3934  (class class class)co 7145  t crest 16684
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321  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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16686
This theorem is referenced by:  firest  16696  restbas  21696  tgrest  21697  resttopon  21699  restcld  21710  restfpw  21717  neitr  21718  restntr  21720  ordtrest  21740  cnrest  21823  lmss  21836  connsubclo  21962  restnlly  22020  islly2  22022  cldllycmp  22033  lly1stc  22034  kgenss  22081  xkococnlem  22197  xkoinjcn  22225  qtoprest  22255  trfbas2  22381  trfil1  22424  trfil2  22425  fgtr  22428  trfg  22429  uzrest  22435  trufil  22448  flimrest  22521  cnextcn  22605  trust  22767  restutop  22775  trcfilu  22832  cfiluweak  22833  xrsmopn  23349  zdis  23353  xrge0tsms  23371  cnheibor  23488  cfilres  23828  lhop2  24541  psercn  24943  xrlimcnp  25474  xrge0tsmsd  30620  ordtrestNEW  31064  pnfneige0  31094  lmxrge0  31095  rrhre  31162  cvmscld  32418  cvmopnlem  32423  cvmliftmolem1  32426  poimirlem30  34804  subspopn  34910  iocopn  41676  icoopn  41681  limcresiooub  41803  limcresioolb  41804  fourierdlem32  42305  fourierdlem33  42306  fourierdlem48  42320  fourierdlem49  42321
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