Theorem resttopon 23279

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 23031	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 23044	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5284	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 608	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23278	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 697	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		sseqin2 4178	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
9	8	bilani 509	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
10		simpl 487	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11	3	adantr 485	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
12		elrestr 17471	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
13	10, 5, 11, 12	syl3anc 1394	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	9, 13	eqeltrrd 2866	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
15		elssuni 4900	. . . 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 17469	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
18	5, 17	syldan 602	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19		inss2 4192	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
20		vex 3461	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21	20	inex1 5278	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
22	21	elpw 4562	. . . . . . . . 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 7100	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
26	25	frnd 6704	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
27	18, 26	eqsstrd 3973	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
28		sspwuni 5062	. . . 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 3956	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
31		istopon 23030	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
32	7, 30, 31	sylanbrc 594	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868   ↦ cmpt 5186  ran crn 5653  ‘cfv 6525  (class class class)co 7400   ↾_t crest 17463  Topctop 23011  TopOnctopon 23028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-en 8932  df-fin 8935  df-fi 9359  df-rest 17465  df-topgen 17486  df-top 23012  df-topon 23029  df-bases 23064

This theorem is referenced by:  restuni  23280  stoig  23281  restsn2  23289  restlp  23301  restperf  23302  perfopn  23303  cnrest  23403  cnrest2  23404  cnrest2r  23405  cnpresti  23406  cnprest  23407  cnprest2  23408  restcnrm  23480  connsuba  23538  kgentopon  23656  1stckgenlem  23671  kgen2ss  23673  kgencn  23674  xkoinjcn  23805  qtoprest  23835  flimrest  24101  fclsrest  24142  flfcntr  24161  efmndtmd  24219  symgtgp  24224  dvrcn  24302  sszcld  24936  divcn  24988  cncfmptc  25032  cncfmptid  25033  cncfmpt2f  25035  cdivcncf  25041  cnmpopc  25048  icchmeo  25061  htpycc  25100  pcocn  25137  pcohtpylem  25139  pcopt  25142  pcopt2  25143  pcoass  25144  pcorevlem  25146  relcmpcmet  25438  mulcncf  25566  limcvallem  25991  ellimc2  25997  limcres  26006  cnplimc  26007  cnlimc  26008  limccnp  26011  limccnp2  26012  dvbss  26021  perfdvf  26023  dvreslem  26029  dvres2lem  26030  dvcnp2  26040  dvcn  26041  dvaddbr  26058  dvmulbr  26059  dvcmulf  26065  dvmptres2  26082  dvmptcmul  26084  dvmptntr  26091  dvmptfsum  26095  dvcnvlem  26096  dvcnv  26097  lhop1lem  26133  lhop2  26135  lhop  26136  dvcnvrelem2  26138  dvcnvre  26139  ftc1lem3  26158  ftc1cn  26163  taylthlem1  26494  ulmdvlem3  26523  psercn  26547  abelth  26562  logcn  26770  cxpcn  26868  cxpcn2  26869  cxpcn3  26871  resqrtcn  26872  sqrtcn  26873  loglesqrt  26884  xrlimcnp  27091  efrlim  27092  ftalem3  27197  xrge0pluscn  34247  xrge0mulc1cn  34248  lmlimxrge0  34255  pnfneige0  34258  lmxrge0  34259  esumcvg  34393  cxpcncf1  34899  cvxpconn  35605  cvxsconn  35606  cvmsf1o  35635  cvmliftlem8  35655  cvmlift2lem9a  35666  cvmlift2lem11  35676  cvmlift3lem6  35687  ivthALT  36708  poimir  38164  broucube  38165  cnambfre  38179  ftc1cnnc  38203  areacirclem2  38220  areacirclem4  38222  fsumcncf  46450  ioccncflimc  46457  cncfuni  46458  icccncfext  46459  icocncflimc  46461  cncfiooicclem1  46465  cxpcncf2  46471  dvmptconst  46487  dvmptidg  46489  dvresntr  46490  itgsubsticclem  46547  dirkercncflem2  46676  dirkercncflem4  46678  fourierdlem32  46711  fourierdlem33  46712  fourierdlem62  46740  fourierdlem93  46771  fourierdlem101  46779