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Theorem elrestr 16029
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 2621 . . . 4 (𝐴𝑆) = (𝐴𝑆)
2 ineq1 3791 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑆) = (𝐴𝑆))
32eqeq2d 2631 . . . . 5 (𝑥 = 𝐴 → ((𝐴𝑆) = (𝑥𝑆) ↔ (𝐴𝑆) = (𝐴𝑆)))
43rspcev 3299 . . . 4 ((𝐴𝐽 ∧ (𝐴𝑆) = (𝐴𝑆)) → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
51, 4mpan2 706 . . 3 (𝐴𝐽 → ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆))
6 elrest 16028 . . 3 ((𝐽𝑉𝑆𝑊) → ((𝐴𝑆) ∈ (𝐽t 𝑆) ↔ ∃𝑥𝐽 (𝐴𝑆) = (𝑥𝑆)))
75, 6syl5ibr 236 . 2 ((𝐽𝑉𝑆𝑊) → (𝐴𝐽 → (𝐴𝑆) ∈ (𝐽t 𝑆)))
873impia 1258 1 ((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2909  cin 3559  (class class class)co 6615  t crest 16021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-rest 16023
This theorem is referenced by:  firest  16033  restbas  20902  tgrest  20903  resttopon  20905  restcld  20916  restfpw  20923  neitr  20924  restntr  20926  ordtrest  20946  cnrest  21029  lmss  21042  connsubclo  21167  restnlly  21225  islly2  21227  cldllycmp  21238  lly1stc  21239  kgenss  21286  xkococnlem  21402  xkoinjcn  21430  qtoprest  21460  trfbas2  21587  trfil1  21630  trfil2  21631  fgtr  21634  trfg  21635  uzrest  21641  trufil  21654  flimrest  21727  cnextcn  21811  trust  21973  restutop  21981  trcfilu  22038  cfiluweak  22039  xrsmopn  22555  zdis  22559  xrge0tsms  22577  cnheibor  22694  cfilres  23034  lhop2  23716  psercn  24118  xrlimcnp  24629  xrge0tsmsd  29612  ordtrestNEW  29791  pnfneige0  29821  lmxrge0  29822  rrhre  29889  cvmscld  31016  cvmopnlem  31021  cvmliftmolem1  31024  poimirlem30  33110  subspopn  33219  iocopn  39192  icoopn  39197  limcresiooub  39310  limcresioolb  39311  fourierdlem32  39693  fourierdlem33  39694  fourierdlem48  39708  fourierdlem49  39709
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