Theorem resttopon 23355

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 23107	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 23120	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5295	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 609	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23354	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 698	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		sseqin2 4179	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
9	8	bilani 510	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
10		simpl 488	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11	3	adantr 486	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
12		elrestr 17506	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
13	10, 5, 11, 12	syl3anc 1398	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	9, 13	eqeltrrd 2867	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
15		elssuni 4909	. . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
16	14, 15	syl 18	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
17		restval 17504	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
18	5, 17	syldan 603	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19		inss2 4193	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
20		vex 3462	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21	20	inex1 5291	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
22	21	elpw 4571	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
23	19, 22	mpbir 234	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
24	23	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
25	24	fmpttd 7117	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
26	25	frnd 6721	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
27	18, 26	eqsstrd 3974	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
28		sspwuni 5071	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
29	27, 28	sylib 221	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
30	16, 29	eqssd 3957	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
31		istopon 23106	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
32	7, 30, 31	sylanbrc 595	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877   ↦ cmpt 5197  ran crn 5667  ‘cfv 6543  (class class class)co 7423   ↾_t crest 17498  Topctop 23087  TopOnctopon 23104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-en 8953  df-fin 8956  df-fi 9381  df-rest 17500  df-topgen 17521  df-top 23088  df-topon 23105  df-bases 23140

This theorem is used by:  restuni  23356  stoig  23357  restsn2  23365  restlp  23377  restperf  23378  perfopn  23379  cnrest  23479  cnrest2  23480  cnrest2r  23481  cnpresti  23482  cnprest  23483  cnprest2  23484  restcnrm  23556  connsuba  23614  kgentopon  23732  1stckgenlem  23747  kgen2ss  23749  kgencn  23750  xkoinjcn  23881  qtoprest  23911  flimrest  24177  fclsrest  24218  flfcntr  24237  efmndtmd  24295  symgtgp  24300  dvrcn  24378  sszcld  25012  divcn  25064  cncfmptc  25108  cncfmptid  25109  cncfmpt2f  25111  cdivcncf  25117  cnmpopc  25124  icchmeo  25137  htpycc  25176  pcocn  25213  pcohtpylem  25215  pcopt  25218  pcopt2  25219  pcoass  25220  pcorevlem  25222  relcmpcmet  25514  mulcncf  25642  limcvallem  26067  ellimc2  26073  limcres  26082  cnplimc  26083  cnlimc  26084  limccnp  26087  limccnp2  26088  dvbss  26097  perfdvf  26099  dvreslem  26105  dvres2lem  26106  dvcnp2  26116  dvcn  26117  dvaddbr  26134  dvmulbr  26135  dvcmulf  26141  dvmptres2  26158  dvmptcmul  26160  dvmptntr  26167  dvmptfsum  26171  dvcnvlem  26172  dvcnv  26173  lhop1lem  26209  lhop2  26211  lhop  26212  dvcnvrelem2  26214  dvcnvre  26215  ftc1lem3  26234  ftc1cn  26239  taylthlem1  26573  ulmdvlem3  26602  psercn  26626  abelth  26641  logcn  26849  cxpcn  26947  cxpcn2  26948  cxpcn3  26950  resqrtcn  26951  sqrtcn  26952  loglesqrt  26963  xrlimcnp  27170  efrlim  27171  ftalem3  27276  xrge0pluscn  34361  xrge0mulc1cn  34362  lmlimxrge0  34369  pnfneige0  34372  lmxrge0  34373  esumcvg  34507  cxpcncf1  35014  cvxpconn  35755  cvxsconn  35756  cvmsf1o  35785  cvmliftlem8  35805  cvmlift2lem9a  35816  cvmlift2lem11  35826  cvmlift3lem6  35837  ivthALT  36887  poimir  38345  broucube  38346  cnambfre  38360  ftc1cnnc  38384  areacirclem2  38401  areacirclem4  38403  fsumcncf  46633  ioccncflimc  46640  cncfuni  46641  icccncfext  46642  icocncflimc  46644  cncfiooicclem1  46648  cxpcncf2  46654  dvmptconst  46670  dvmptidg  46672  dvresntr  46673  itgsubsticclem  46730  dirkercncflem2  46859  dirkercncflem4  46861  fourierdlem32  46894  fourierdlem33  46895  fourierdlem62  46923  fourierdlem93  46954  fourierdlem101  46962