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Theorem elrestr 14918
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 2454 . . . 4  |-  ( A  i^i  S )  =  ( A  i^i  S
)
2 ineq1 3679 . . . . . 6  |-  ( x  =  A  ->  (
x  i^i  S )  =  ( A  i^i  S ) )
32eqeq2d 2468 . . . . 5  |-  ( x  =  A  ->  (
( A  i^i  S
)  =  ( x  i^i  S )  <->  ( A  i^i  S )  =  ( A  i^i  S ) ) )
43rspcev 3207 . . . 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 669 . . 3  |-  ( A  e.  J  ->  E. x  e.  J  ( A  i^i  S )  =  ( x  i^i  S ) )
6 elrest 14917 . . 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 221 . 2  |-  ( ( J  e.  V  /\  S  e.  W )  ->  ( A  e.  J  ->  ( A  i^i  S
)  e.  ( Jt  S ) ) )
873impia 1191 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    i^i cin 3460  (class class class)co 6270   ↾t crest 14910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-rest 14912
This theorem is referenced by:  firest  14922  restbas  19826  tgrest  19827  resttopon  19829  restcld  19840  restfpw  19847  neitr  19848  restntr  19850  ordtrest  19870  cnrest  19953  lmss  19966  consubclo  20091  restnlly  20149  islly2  20151  cldllycmp  20162  lly1stc  20163  kgenss  20210  xkococnlem  20326  xkoinjcn  20354  qtoprest  20384  trfbas2  20510  trfil1  20553  trfil2  20554  fgtr  20557  trfg  20558  uzrest  20564  trufil  20577  flimrest  20650  cnextcn  20733  trust  20898  restutop  20906  trcfilu  20963  cfiluweak  20964  xrsmopn  21483  zdis  21487  xrge0tsms  21505  cnheibor  21621  cfilres  21901  lhop2  22582  psercn  22987  xrlimcnp  23496  xrge0tsmsd  28010  ordtrestNEW  28138  pnfneige0  28168  lmxrge0  28169  rrhre  28233  cvmscld  28982  cvmopnlem  28987  cvmliftmolem1  28990  subspopn  30485  iocopn  31799  icoopn  31804  limcresiooub  31887  limcresioolb  31888  fourierdlem32  32160  fourierdlem33  32161  fourierdlem48  32176  fourierdlem49  32177
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