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Theorem elrestr 15375
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  |-  ( ( J  e.  V  /\  S  e.  W  /\  A  e.  J )  ->  ( A  i^i  S
)  e.  ( Jt  S ) )

Proof of Theorem elrestr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2461 . . . 4  |-  ( A  i^i  S )  =  ( A  i^i  S
)
2 ineq1 3638 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  S )  =  ( A  i^i  S ) )
32eqeq2d 2471 . . . . 5  |-  ( x  =  A  ->  (
( A  i^i  S
)  =  ( x  i^i  S )  <->  ( A  i^i  S )  =  ( A  i^i  S ) ) )
43rspcev 3161 . . . 4  |-  ( ( A  e.  J  /\  ( A  i^i  S )  =  ( A  i^i  S ) )  ->  E. x  e.  J  ( A  i^i  S )  =  ( x  i^i  S ) )
51, 4mpan2 682 . . 3  |-  ( A  e.  J  ->  E. x  e.  J  ( A  i^i  S )  =  ( x  i^i  S ) )
6 elrest 15374 . . 3  |-  ( ( J  e.  V  /\  S  e.  W )  ->  ( ( A  i^i  S )  e.  ( Jt  S )  <->  E. x  e.  J  ( A  i^i  S )  =  ( x  i^i 
S ) ) )
75, 6syl5ibr 229 . 2  |-  ( ( J  e.  V  /\  S  e.  W )  ->  ( A  e.  J  ->  ( A  i^i  S
)  e.  ( Jt  S ) ) )
873impia 1212 1  |-  ( ( J  e.  V  /\  S  e.  W  /\  A  e.  J )  ->  ( A  i^i  S
)  e.  ( Jt  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   E.wrex 2749    i^i cin 3414  (class class class)co 6314   ↾t crest 15367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-rest 15369
This theorem is referenced by:  firest  15379  restbas  20222  tgrest  20223  resttopon  20225  restcld  20236  restfpw  20243  neitr  20244  restntr  20246  ordtrest  20266  cnrest  20349  lmss  20362  consubclo  20487  restnlly  20545  islly2  20547  cldllycmp  20558  lly1stc  20559  kgenss  20606  xkococnlem  20722  xkoinjcn  20750  qtoprest  20780  trfbas2  20906  trfil1  20949  trfil2  20950  fgtr  20953  trfg  20954  uzrest  20960  trufil  20973  flimrest  21046  cnextcn  21130  trust  21292  restutop  21300  trcfilu  21357  cfiluweak  21358  xrsmopn  21878  zdis  21882  xrge0tsms  21900  cnheibor  22031  cfilres  22314  lhop2  23015  psercn  23429  xrlimcnp  23942  xrge0tsmsd  28596  ordtrestNEW  28775  pnfneige0  28805  lmxrge0  28806  rrhre  28873  cvmscld  30044  cvmopnlem  30049  cvmliftmolem1  30052  poimirlem30  32014  subspopn  32125  iocopn  37658  icoopn  37663  limcresiooub  37760  limcresioolb  37761  fourierdlem32  38039  fourierdlem33  38040  fourierdlem48  38055  fourierdlem49  38056
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