Theorem resttopon 23329

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 23081	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 23094	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5289	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 608	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23328	. . 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 4175	. . . . . 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 17487	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
13	10, 5, 11, 12	syl3anc 1397	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	9, 13	eqeltrrd 2863	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
15		elssuni 4903	. . . 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 17485	. . . . . 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 4189	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
20		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21	20	inex1 5285	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
22	21	elpw 4565	. . . . . . . . 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 7110	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
26	25	frnd 6714	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
27	18, 26	eqsstrd 3970	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
28		sspwuni 5065	. . . 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 3953	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
31		istopon 23080	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871   ↦ cmpt 5191  ran crn 5661  ‘cfv 6536  (class class class)co 7412   ↾_t crest 17479  Topctop 23061  TopOnctopon 23078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-en 8942  df-fin 8945  df-fi 9369  df-rest 17481  df-topgen 17502  df-top 23062  df-topon 23079  df-bases 23114

This theorem is used by:  restuni  23330  stoig  23331  restsn2  23339  restlp  23351  restperf  23352  perfopn  23353  cnrest  23453  cnrest2  23454  cnrest2r  23455  cnpresti  23456  cnprest  23457  cnprest2  23458  restcnrm  23530  connsuba  23588  kgentopon  23706  1stckgenlem  23721  kgen2ss  23723  kgencn  23724  xkoinjcn  23855  qtoprest  23885  flimrest  24151  fclsrest  24192  flfcntr  24211  efmndtmd  24269  symgtgp  24274  dvrcn  24352  sszcld  24986  divcn  25038  cncfmptc  25082  cncfmptid  25083  cncfmpt2f  25085  cdivcncf  25091  cnmpopc  25098  icchmeo  25111  htpycc  25150  pcocn  25187  pcohtpylem  25189  pcopt  25192  pcopt2  25193  pcoass  25194  pcorevlem  25196  relcmpcmet  25488  mulcncf  25616  limcvallem  26041  ellimc2  26047  limcres  26056  cnplimc  26057  cnlimc  26058  limccnp  26061  limccnp2  26062  dvbss  26071  perfdvf  26073  dvreslem  26079  dvres2lem  26080  dvcnp2  26090  dvcn  26091  dvaddbr  26108  dvmulbr  26109  dvcmulf  26115  dvmptres2  26132  dvmptcmul  26134  dvmptntr  26141  dvmptfsum  26145  dvcnvlem  26146  dvcnv  26147  lhop1lem  26183  lhop2  26185  lhop  26186  dvcnvrelem2  26188  dvcnvre  26189  ftc1lem3  26208  ftc1cn  26213  taylthlem1  26547  ulmdvlem3  26576  psercn  26600  abelth  26615  logcn  26823  cxpcn  26921  cxpcn2  26922  cxpcn3  26924  resqrtcn  26925  sqrtcn  26926  loglesqrt  26937  xrlimcnp  27144  efrlim  27145  ftalem3  27250  xrge0pluscn  34339  xrge0mulc1cn  34340  lmlimxrge0  34347  pnfneige0  34350  lmxrge0  34351  esumcvg  34485  cxpcncf1  34991  cvxpconn  35742  cvxsconn  35743  cvmsf1o  35772  cvmliftlem8  35792  cvmlift2lem9a  35803  cvmlift2lem11  35813  cvmlift3lem6  35824  ivthALT  36874  poimir  38332  broucube  38333  cnambfre  38347  ftc1cnnc  38371  areacirclem2  38388  areacirclem4  38390  fsumcncf  46620  ioccncflimc  46627  cncfuni  46628  icccncfext  46629  icocncflimc  46631  cncfiooicclem1  46635  cxpcncf2  46641  dvmptconst  46657  dvmptidg  46659  dvresntr  46660  itgsubsticclem  46717  dirkercncflem2  46846  dirkercncflem4  46848  fourierdlem32  46881  fourierdlem33  46882  fourierdlem62  46910  fourierdlem93  46941  fourierdlem101  46949