MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resttopon Structured version   Unicode version

Theorem resttopon 19468
Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
resttopon  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )

Proof of Theorem resttopon
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 topontop 19234 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  J  e.  Top )
3 id 22 . . . 4  |-  ( A 
C_  X  ->  A  C_  X )
4 toponmax 19236 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 ssexg 4593 . . . 4  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
63, 4, 5syl2anr 478 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
7 resttop 19467 . . 3  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( Jt  A )  e.  Top )
82, 6, 7syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  Top )
9 simpr 461 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
10 dfss1 3703 . . . . . 6  |-  ( A 
C_  X  <->  ( X  i^i  A )  =  A )
119, 10sylib 196 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( X  i^i  A )  =  A )
12 simpl 457 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  J  e.  (TopOn `  X )
)
134adantr 465 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
14 elrestr 14687 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V  /\  X  e.  J )  ->  ( X  i^i  A )  e.  ( Jt  A ) )
1512, 6, 13, 14syl3anc 1228 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( X  i^i  A )  e.  ( Jt  A ) )
1611, 15eqeltrrd 2556 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  ( Jt  A ) )
17 elssuni 4275 . . . 4  |-  ( A  e.  ( Jt  A )  ->  A  C_  U. ( Jt  A ) )
1816, 17syl 16 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_ 
U. ( Jt  A ) )
19 restval 14685 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
206, 19syldan 470 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
21 inss2 3719 . . . . . . . . 9  |-  ( x  i^i  A )  C_  A
22 vex 3116 . . . . . . . . . . 11  |-  x  e. 
_V
2322inex1 4588 . . . . . . . . . 10  |-  ( x  i^i  A )  e. 
_V
2423elpw 4016 . . . . . . . . 9  |-  ( ( x  i^i  A )  e.  ~P A  <->  ( x  i^i  A )  C_  A
)
2521, 24mpbir 209 . . . . . . . 8  |-  ( x  i^i  A )  e. 
~P A
2625a1i 11 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  x  e.  J )  ->  (
x  i^i  A )  e.  ~P A )
27 eqid 2467 . . . . . . 7  |-  ( x  e.  J  |->  ( x  i^i  A ) )  =  ( x  e.  J  |->  ( x  i^i 
A ) )
2826, 27fmptd 6046 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
x  e.  J  |->  ( x  i^i  A ) ) : J --> ~P A
)
29 frn 5737 . . . . . 6  |-  ( ( x  e.  J  |->  ( x  i^i  A ) ) : J --> ~P A  ->  ran  ( x  e.  J  |->  ( x  i^i 
A ) )  C_  ~P A )
3028, 29syl 16 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  C_  ~P A )
3120, 30eqsstrd 3538 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  C_  ~P A )
32 sspwuni 4411 . . . 4  |-  ( ( Jt  A )  C_  ~P A 
<-> 
U. ( Jt  A ) 
C_  A )
3331, 32sylib 196 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  U. ( Jt  A )  C_  A
)
3418, 33eqssd 3521 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
35 istopon 19233 . 2  |-  ( ( Jt  A )  e.  (TopOn `  A )  <->  ( ( Jt  A )  e.  Top  /\  A  =  U. ( Jt  A ) ) )
368, 34, 35sylanbrc 664 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    |-> cmpt 4505   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6285   ↾t crest 14679   Topctop 19201  TopOnctopon 19202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-oadd 7135  df-er 7312  df-en 7518  df-fin 7521  df-fi 7872  df-rest 14681  df-topgen 14702  df-top 19206  df-bases 19208  df-topon 19209
This theorem is referenced by:  restuni  19469  stoig  19470  restsn2  19478  restlp  19490  restperf  19491  perfopn  19492  cnrest  19592  cnrest2  19593  cnrest2r  19594  cnpresti  19595  cnprest  19596  cnprest2  19597  restcnrm  19669  consuba  19727  kgentopon  19866  1stckgenlem  19881  kgen2ss  19883  kgencn  19884  xkoinjcn  20015  qtoprest  20045  flimrest  20311  fclsrest  20352  symgtgp  20427  dvrcn  20513  sszcld  21149  divcn  21199  cncfmptc  21242  cncfmptid  21243  cncfmpt2f  21245  cdivcncf  21248  cnmpt2pc  21255  icchmeo  21268  htpycc  21307  pcocn  21344  pcohtpylem  21346  pcopt  21349  pcopt2  21350  pcoass  21351  pcorevlem  21353  relcmpcmet  21582  limcvallem  22102  ellimc2  22108  limcres  22117  cnplimc  22118  cnlimc  22119  limccnp  22122  limccnp2  22123  dvbss  22132  perfdvf  22134  dvreslem  22140  dvres2lem  22141  dvcnp2  22150  dvcn  22151  dvaddbr  22168  dvmulbr  22169  dvcmulf  22175  dvmptres2  22192  dvmptcmul  22194  dvmptntr  22201  dvmptfsum  22203  dvcnvlem  22204  dvcnv  22205  lhop1lem  22241  lhop2  22243  lhop  22244  dvcnvrelem2  22246  dvcnvre  22247  ftc1lem3  22266  ftc1cn  22271  taylthlem1  22594  ulmdvlem3  22623  psercn  22647  abelth  22662  logcn  22853  cxpcn  22944  cxpcn2  22945  cxpcn3  22947  resqrtcn  22948  sqrtcn  22949  loglesqrt  22957  xrlimcnp  23123  efrlim  23124  ftalem3  23173  xrge0pluscn  27673  xrge0mulc1cn  27674  lmlimxrge0  27681  pnfneige0  27684  lmxrge0  27685  esumcvg  27843  cvxpcon  28438  cvxscon  28439  cvmsf1o  28468  cvmliftlem8  28488  cvmlift2lem9a  28499  cvmlift2lem11  28509  cvmlift3lem6  28520  cnambfre  29916  ftc1cnnc  29942  areacirclem2  29961  areacirclem4  29963  ivthALT  30006  fsumcncf  31443  ioccncflimc  31451  cncfuni  31452  icccncfext  31453  icocncflimc  31455  cncfiooicclem1  31459  dvmptconst  31470  dvmptidg  31472  dvresntr  31473  itgsubsticclem  31520  dirkercncflem2  31631  dirkercncflem4  31633  fourierdlem32  31666  fourierdlem33  31667  fourierdlem62  31696  fourierdlem93  31727  fourierdlem101  31735
  Copyright terms: Public domain W3C validator