Theorem elrestr 16089

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. ( J ↾t S ) )

Proof of Theorem elrestr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( A i^i S ) = ( A i^i S )
2		ineq1 3807	. . . . . 6 |- ( x = A -> ( x i^i S ) = ( A i^i S ) )
3	2	eqeq2d 2632	. . . . 5 |- ( x = A -> ( ( A i^i S ) = ( x i^i S ) <-> ( A i^i S ) = ( A i^i S ) ) )
4	3	rspcev 3309	. . . 4 |- ( ( A e. J /\ ( A i^i S ) = ( A i^i S ) ) -> E. x e. J ( A i^i S ) = ( x i^i S ) )
5	1, 4	mpan2 707	. . 3 |- ( A e. J -> E. x e. J ( A i^i S ) = ( x i^i S ) )
6		elrest 16088	. . 3 |- ( ( J e. V /\ S e. W ) -> ( ( A i^i S ) e. ( J ↾t S ) <-> E. x e. J ( A i^i S ) = ( x i^i S ) ) )
7	5, 6	syl5ibr 236	. 2 |- ( ( J e. V /\ S e. W ) -> ( A e. J -> ( A i^i S ) e. ( J ↾t S ) ) )
8	7	3impia 1261	1 |- ( ( J e. V /\ S e. W /\ A e. J ) -> ( A i^i S ) e. ( J ↾t S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573  (class class class)co 6650   ↾_t crest 16081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by:  firest  16093  restbas  20962  tgrest  20963  resttopon  20965  restcld  20976  restfpw  20983  neitr  20984  restntr  20986  ordtrest  21006  cnrest  21089  lmss  21102  connsubclo  21227  restnlly  21285  islly2  21287  cldllycmp  21298  lly1stc  21299  kgenss  21346  xkococnlem  21462  xkoinjcn  21490  qtoprest  21520  trfbas2  21647  trfil1  21690  trfil2  21691  fgtr  21694  trfg  21695  uzrest  21701  trufil  21714  flimrest  21787  cnextcn  21871  trust  22033  restutop  22041  trcfilu  22098  cfiluweak  22099  xrsmopn  22615  zdis  22619  xrge0tsms  22637  cnheibor  22754  cfilres  23094  lhop2  23778  psercn  24180  xrlimcnp  24695  xrge0tsmsd  29785  ordtrestNEW  29967  pnfneige0  29997  lmxrge0  29998  rrhre  30065  cvmscld  31255  cvmopnlem  31260  cvmliftmolem1  31263  poimirlem30  33439  subspopn  33548  iocopn  39746  icoopn  39751  limcresiooub  39874  limcresioolb  39875  fourierdlem32  40356  fourierdlem33  40357  fourierdlem48  40371  fourierdlem49  40372