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Theorem elrestr 16449
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 2825 . . . 4 (𝐴𝑆) = (𝐴𝑆)
2 ineq1 4036 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑆) = (𝐴𝑆))
32rspceeqv 3544 . . . 4 ((𝐴𝐽 ∧ (𝐴𝑆) = (𝐴𝑆)) → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
41, 3mpan2 682 . . 3 (𝐴𝐽 → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
5 elrest 16448 . . 3 ((𝐽𝑉𝑆𝑊) → ((𝐴𝑆) ∈ (𝐽t 𝑆) ↔ ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆)))
64, 5syl5ibr 238 . 2 ((𝐽𝑉𝑆𝑊) → (𝐴𝐽 → (𝐴𝑆) ∈ (𝐽t 𝑆)))
763impia 1149 1 ((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wrex 3118  cin 3797  (class class class)co 6910  t crest 16441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-rest 16443
This theorem is referenced by:  firest  16453  restbas  21340  tgrest  21341  resttopon  21343  restcld  21354  restfpw  21361  neitr  21362  restntr  21364  ordtrest  21384  cnrest  21467  lmss  21480  connsubclo  21605  restnlly  21663  islly2  21665  cldllycmp  21676  lly1stc  21677  kgenss  21724  xkococnlem  21840  xkoinjcn  21868  qtoprest  21898  trfbas2  22024  trfil1  22067  trfil2  22068  fgtr  22071  trfg  22072  uzrest  22078  trufil  22091  flimrest  22164  cnextcn  22248  trust  22410  restutop  22418  trcfilu  22475  cfiluweak  22476  xrsmopn  22992  zdis  22996  xrge0tsms  23014  cnheibor  23131  cfilres  23471  lhop2  24184  psercn  24586  xrlimcnp  25115  xrge0tsmsd  30326  ordtrestNEW  30508  pnfneige0  30538  lmxrge0  30539  rrhre  30606  cvmscld  31797  cvmopnlem  31802  cvmliftmolem1  31805  poimirlem30  33978  subspopn  34085  iocopn  40536  icoopn  40541  limcresiooub  40663  limcresioolb  40664  fourierdlem32  41144  fourierdlem33  41145  fourierdlem48  41159  fourierdlem49  41160
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