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Theorem elrestr 16697
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 2826 . . . 4 (𝐴𝑆) = (𝐴𝑆)
2 ineq1 4185 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑆) = (𝐴𝑆))
32rspceeqv 3642 . . . 4 ((𝐴𝐽 ∧ (𝐴𝑆) = (𝐴𝑆)) → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
41, 3mpan2 687 . . 3 (𝐴𝐽 → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
5 elrest 16696 . . 3 ((𝐽𝑉𝑆𝑊) → ((𝐴𝑆) ∈ (𝐽t 𝑆) ↔ ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆)))
64, 5syl5ibr 247 . 2 ((𝐽𝑉𝑆𝑊) → (𝐴𝐽 → (𝐴𝑆) ∈ (𝐽t 𝑆)))
763impia 1111 1 ((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wrex 3144  cin 3939  (class class class)co 7150  t crest 16689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rest 16691
This theorem is referenced by:  firest  16701  restbas  21701  tgrest  21702  resttopon  21704  restcld  21715  restfpw  21722  neitr  21723  restntr  21725  ordtrest  21745  cnrest  21828  lmss  21841  connsubclo  21967  restnlly  22025  islly2  22027  cldllycmp  22038  lly1stc  22039  kgenss  22086  xkococnlem  22202  xkoinjcn  22230  qtoprest  22260  trfbas2  22386  trfil1  22429  trfil2  22430  fgtr  22433  trfg  22434  uzrest  22440  trufil  22453  flimrest  22526  cnextcn  22610  trust  22772  restutop  22780  trcfilu  22837  cfiluweak  22838  xrsmopn  23354  zdis  23358  xrge0tsms  23376  cnheibor  23493  cfilres  23833  lhop2  24546  psercn  24948  xrlimcnp  25479  xrge0tsmsd  30625  ordtrestNEW  31069  pnfneige0  31099  lmxrge0  31100  rrhre  31167  cvmscld  32423  cvmopnlem  32428  cvmliftmolem1  32431  poimirlem30  34808  subspopn  34914  iocopn  41680  icoopn  41685  limcresiooub  41807  limcresioolb  41808  fourierdlem32  42309  fourierdlem33  42310  fourierdlem48  42324  fourierdlem49  42325
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