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Theorem resttopon 18724
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 18490 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 462 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  J  e.  Top )
3 id 22 . . . 4  |-  ( A 
C_  X  ->  A  C_  X )
4 toponmax 18492 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 ssexg 4435 . . . 4  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
63, 4, 5syl2anr 475 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
7 resttop 18723 . . 3  |-  ( ( J  e.  Top  /\  A  e.  _V )  ->  ( Jt  A )  e.  Top )
82, 6, 7syl2anc 656 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  Top )
9 simpr 458 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
10 dfss1 3552 . . . . . 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 454 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  J  e.  (TopOn `  X )
)
134adantr 462 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
14 elrestr 14363 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V  /\  X  e.  J )  ->  ( X  i^i  A )  e.  ( Jt  A ) )
1512, 6, 13, 14syl3anc 1213 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( X  i^i  A )  e.  ( Jt  A ) )
1611, 15eqeltrrd 2516 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  ( Jt  A ) )
17 elssuni 4118 . . . 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 14361 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
206, 19syldan 467 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
21 inss2 3568 . . . . . . . . 9  |-  ( x  i^i  A )  C_  A
22 vex 2973 . . . . . . . . . . 11  |-  x  e. 
_V
2322inex1 4430 . . . . . . . . . 10  |-  ( x  i^i  A )  e. 
_V
2423elpw 3863 . . . . . . . . 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 2441 . . . . . . 7  |-  ( x  e.  J  |->  ( x  i^i  A ) )  =  ( x  e.  J  |->  ( x  i^i 
A ) )
2826, 27fmptd 5864 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
x  e.  J  |->  ( x  i^i  A ) ) : J --> ~P A
)
29 frn 5562 . . . . . 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 3387 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  C_  ~P A )
32 sspwuni 4253 . . . 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 3370 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  =  U. ( Jt  A ) )
35 istopon 18489 . 2  |-  ( ( Jt  A )  e.  (TopOn `  A )  <->  ( ( Jt  A )  e.  Top  /\  A  =  U. ( Jt  A ) ) )
368, 34, 35sylanbrc 659 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 1364    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   U.cuni 4088    e. cmpt 4347   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090   ↾t crest 14355   Topctop 18457  TopOnctopon 18458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-oadd 6920  df-er 7097  df-en 7307  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465
This theorem is referenced by:  restuni  18725  stoig  18726  restsn2  18734  restlp  18746  restperf  18747  perfopn  18748  cnrest  18848  cnrest2  18849  cnrest2r  18850  cnpresti  18851  cnprest  18852  cnprest2  18853  restcnrm  18925  consuba  18983  kgentopon  19070  1stckgenlem  19085  kgen2ss  19087  kgencn  19088  xkoinjcn  19219  qtoprest  19249  flimrest  19515  fclsrest  19556  symgtgp  19631  dvrcn  19717  sszcld  20353  divcn  20403  cncfmptc  20446  cncfmptid  20447  cncfmpt2f  20449  cdivcncf  20452  cnmpt2pc  20459  icchmeo  20472  htpycc  20511  pcocn  20548  pcohtpylem  20550  pcopt  20553  pcopt2  20554  pcoass  20555  pcorevlem  20557  relcmpcmet  20786  limcvallem  21305  ellimc2  21311  limcres  21320  cnplimc  21321  cnlimc  21322  limccnp  21325  limccnp2  21326  dvbss  21335  perfdvf  21337  dvreslem  21343  dvres2lem  21344  dvcnp2  21353  dvcn  21354  dvaddbr  21371  dvmulbr  21372  dvcmulf  21378  dvmptres2  21395  dvmptcmul  21397  dvmptntr  21404  dvmptfsum  21406  dvcnvlem  21407  dvcnv  21408  lhop1lem  21444  lhop2  21446  lhop  21447  dvcnvrelem2  21449  dvcnvre  21450  ftc1lem3  21469  ftc1cn  21474  taylthlem1  21797  ulmdvlem3  21826  psercn  21850  abelth  21865  logcn  22051  cxpcn  22142  cxpcn2  22143  cxpcn3  22145  resqrcn  22146  sqrcn  22147  loglesqr  22155  xrlimcnp  22321  efrlim  22322  ftalem3  22371  xrge0pluscn  26306  xrge0mulc1cn  26307  lmlimxrge0  26314  pnfneige0  26317  lmxrge0  26318  esumcvg  26471  cvxpcon  27061  cvxscon  27062  cvmsf1o  27091  cvmliftlem8  27111  cvmlift2lem9a  27122  cvmlift2lem11  27132  cvmlift3lem6  27143  cnambfre  28365  ftc1cnnc  28391  areacirclem2  28410  areacirclem4  28412  ivthALT  28455
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