Theorem resttopon 23082

Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
resttopon	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Proof of Theorem resttopon

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 22834	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 22847	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5263	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 597	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23081	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 685	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		simpr 484	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
9		sseqin2 4172	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
10	8, 9	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
11		simpl 482	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
12	3	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
13		elrestr 17338	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	11, 5, 12, 13	syl3anc 1373	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	10, 14	eqeltrrd 2832	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
16		elssuni 4889	. . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
17	15, 16	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
18		restval 17336	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19	5, 18	syldan 591	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20		inss2 4187	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
21		vex 3440	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
22	21	inex1 5257	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
23	22	elpw 4553	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
24	20, 23	mpbir 231	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
25	24	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
26	25	fmpttd 7054	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
27	26	frnd 6665	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
28	19, 27	eqsstrd 3964	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
29		sspwuni 5050	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
30	28, 29	sylib 218	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
31	17, 30	eqssd 3947	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
32		istopon 22833	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
33	7, 31, 32	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4549  ∪ cuni 4858   ↦ cmpt 5174  ran crn 5620  ‘cfv 6487  (class class class)co 7352   ↾_t crest 17330  Topctop 22814  TopOnctopon 22831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-en 8876  df-fin 8879  df-fi 9301  df-rest 17332  df-topgen 17353  df-top 22815  df-topon 22832  df-bases 22867

This theorem is referenced by:  restuni  23083  stoig  23084  restsn2  23092  restlp  23104  restperf  23105  perfopn  23106  cnrest  23206  cnrest2  23207  cnrest2r  23208  cnpresti  23209  cnprest  23210  cnprest2  23211  restcnrm  23283  connsuba  23341  kgentopon  23459  1stckgenlem  23474  kgen2ss  23476  kgencn  23477  xkoinjcn  23608  qtoprest  23638  flimrest  23904  fclsrest  23945  flfcntr  23964  efmndtmd  24022  symgtgp  24027  dvrcn  24105  sszcld  24739  divcnOLD  24790  divcn  24792  cncfmptc  24838  cncfmptid  24839  cncfmpt2f  24841  cdivcncf  24847  cnmpopc  24855  icchmeo  24871  icchmeoOLD  24872  htpycc  24912  pcocn  24950  pcohtpylem  24952  pcopt  24955  pcopt2  24956  pcoass  24957  pcorevlem  24959  relcmpcmet  25251  mulcncf  25379  limcvallem  25805  ellimc2  25811  limcres  25820  cnplimc  25821  cnlimc  25822  limccnp  25825  limccnp2  25826  dvbss  25835  perfdvf  25837  dvreslem  25843  dvres2lem  25844  dvcnp2  25854  dvcnp2OLD  25855  dvcn  25856  dvaddbr  25873  dvmulbr  25874  dvmulbrOLD  25875  dvcmulf  25881  dvmptres2  25899  dvmptcmul  25901  dvmptntr  25908  dvmptfsum  25912  dvcnvlem  25913  dvcnv  25914  lhop1lem  25951  lhop2  25953  lhop  25954  dvcnvrelem2  25956  dvcnvre  25957  ftc1lem3  25978  ftc1cn  25983  taylthlem1  26314  ulmdvlem3  26344  psercn  26369  abelth  26384  logcn  26589  cxpcn  26687  cxpcnOLD  26688  cxpcn2  26689  cxpcn3  26691  resqrtcn  26692  sqrtcn  26693  loglesqrt  26704  xrlimcnp  26911  efrlim  26912  efrlimOLD  26913  ftalem3  27018  xrge0pluscn  33960  xrge0mulc1cn  33961  lmlimxrge0  33968  pnfneige0  33971  lmxrge0  33972  esumcvg  34106  cxpcncf1  34615  cvxpconn  35293  cvxsconn  35294  cvmsf1o  35323  cvmliftlem8  35343  cvmlift2lem9a  35354  cvmlift2lem11  35364  cvmlift3lem6  35375  ivthALT  36386  poimir  37699  broucube  37700  cnambfre  37714  ftc1cnnc  37738  areacirclem2  37755  areacirclem4  37757  fsumcncf  45981  ioccncflimc  45988  cncfuni  45989  icccncfext  45990  icocncflimc  45992  cncfiooicclem1  45996  cxpcncf2  46002  dvmptconst  46018  dvmptidg  46020  dvresntr  46021  itgsubsticclem  46078  dirkercncflem2  46207  dirkercncflem4  46209  fourierdlem32  46242  fourierdlem33  46243  fourierdlem62  46271  fourierdlem93  46302  fourierdlem101  46310