Theorem resttopon 22220

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 21970	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 21983	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5242	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 596	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 22219	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 681	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		simpr 484	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
9		sseqin2 4146	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
10	8, 9	sylib 217	. . . . 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 17056	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	11, 5, 12, 13	syl3anc 1369	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	10, 14	eqeltrrd 2840	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
16		elssuni 4868	. . . 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 17054	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19	5, 18	syldan 590	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20		inss2 4160	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
21		vex 3426	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
22	21	inex1 5236	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
23	22	elpw 4534	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
24	20, 23	mpbir 230	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
25	24	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
26	25	fmpttd 6971	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
27	26	frnd 6592	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
28	19, 27	eqsstrd 3955	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
29		sspwuni 5025	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
30	28, 29	sylib 217	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
31	17, 30	eqssd 3934	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
32		istopon 21969	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
33	7, 31, 32	sylanbrc 582	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  TopOnctopon 21967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004

This theorem is referenced by:  restuni  22221  stoig  22222  restsn2  22230  restlp  22242  restperf  22243  perfopn  22244  cnrest  22344  cnrest2  22345  cnrest2r  22346  cnpresti  22347  cnprest  22348  cnprest2  22349  restcnrm  22421  connsuba  22479  kgentopon  22597  1stckgenlem  22612  kgen2ss  22614  kgencn  22615  xkoinjcn  22746  qtoprest  22776  flimrest  23042  fclsrest  23083  flfcntr  23102  efmndtmd  23160  symgtgp  23165  dvrcn  23243  sszcld  23886  divcn  23937  cncfmptc  23981  cncfmptid  23982  cncfmpt2f  23984  cdivcncf  23990  cnmpopc  23997  icchmeo  24010  htpycc  24049  pcocn  24086  pcohtpylem  24088  pcopt  24091  pcopt2  24092  pcoass  24093  pcorevlem  24095  relcmpcmet  24387  limcvallem  24940  ellimc2  24946  limcres  24955  cnplimc  24956  cnlimc  24957  limccnp  24960  limccnp2  24961  dvbss  24970  perfdvf  24972  dvreslem  24978  dvres2lem  24979  dvcnp2  24989  dvcn  24990  dvaddbr  25007  dvmulbr  25008  dvcmulf  25014  dvmptres2  25031  dvmptcmul  25033  dvmptntr  25040  dvmptfsum  25044  dvcnvlem  25045  dvcnv  25046  lhop1lem  25082  lhop2  25084  lhop  25085  dvcnvrelem2  25087  dvcnvre  25088  ftc1lem3  25107  ftc1cn  25112  taylthlem1  25437  ulmdvlem3  25466  psercn  25490  abelth  25505  logcn  25707  cxpcn  25803  cxpcn2  25804  cxpcn3  25806  resqrtcn  25807  sqrtcn  25808  loglesqrt  25816  xrlimcnp  26023  efrlim  26024  ftalem3  26129  xrge0pluscn  31792  xrge0mulc1cn  31793  lmlimxrge0  31800  pnfneige0  31803  lmxrge0  31804  esumcvg  31954  cxpcncf1  32475  cvxpconn  33104  cvxsconn  33105  cvmsf1o  33134  cvmliftlem8  33154  cvmlift2lem9a  33165  cvmlift2lem11  33175  cvmlift3lem6  33186  ivthALT  34451  poimir  35737  broucube  35738  cnambfre  35752  ftc1cnnc  35776  areacirclem2  35793  areacirclem4  35795  fsumcncf  43309  ioccncflimc  43316  cncfuni  43317  icccncfext  43318  icocncflimc  43320  cncfiooicclem1  43324  cxpcncf2  43330  dvmptconst  43346  dvmptidg  43348  dvresntr  43349  itgsubsticclem  43406  dirkercncflem2  43535  dirkercncflem4  43537  fourierdlem32  43570  fourierdlem33  43571  fourierdlem62  43599  fourierdlem93  43630  fourierdlem101  43638