Theorem resttopon 23392

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 23144	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 23157	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5288	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 609	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23391	. . 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 4172	. . . . . 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 17519	. . . . . 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 2863	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
15		elssuni 4902	. . . 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 17517	. . . . . 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 4186	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
20		vex 3457	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21	20	inex1 5284	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
22	21	elpw 4564	. . . . . . . . 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 7112	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
26	25	frnd 6715	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
27	18, 26	eqsstrd 3968	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
28		sspwuni 5064	. . . 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 3951	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
31		istopon 23143	. 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 2145  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5190  ran crn 5660  ‘cfv 6537  (class class class)co 7417   ↾_t crest 17511  Topctop 23124  TopOnctopon 23141

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-en 8957  df-fin 8960  df-fi 9385  df-rest 17513  df-topgen 17534  df-top 23125  df-topon 23142  df-bases 23177

This theorem is used by:  restuni  23393  stoig  23394  restsn2  23402  restlp  23414  restperf  23415  perfopn  23416  cnrest  23516  cnrest2  23517  cnrest2r  23518  cnpresti  23519  cnprest  23520  cnprest2  23521  restcnrm  23593  connsuba  23651  kgentopon  23770  1stckgenlem  23785  kgen2ss  23787  kgencn  23788  xkoinjcn  23919  qtoprest  23949  flimrest  24215  fclsrest  24256  flfcntr  24275  efmndtmd  24333  symgtgp  24338  dvrcn  24416  sszcld  25050  divcn  25102  cncfmptc  25146  cncfmptid  25147  cncfmpt2f  25149  cdivcncf  25155  cnmpopc  25162  icchmeo  25175  htpycc  25214  pcocn  25251  pcohtpylem  25253  pcopt  25256  pcopt2  25257  pcoass  25258  pcorevlem  25260  relcmpcmet  25552  mulcncf  25680  limcvallem  26105  ellimc2  26111  limcres  26120  cnplimc  26121  cnlimc  26122  limccnp  26125  limccnp2  26126  dvbss  26135  perfdvf  26137  dvreslem  26143  dvres2lem  26144  dvcnp2  26154  dvcn  26155  dvaddbr  26172  dvmulbr  26173  dvcmulf  26179  dvmptres2  26196  dvmptcmul  26198  dvmptntr  26205  dvmptfsum  26209  dvcnvlem  26210  dvcnv  26211  lhop1lem  26247  lhop2  26249  lhop  26250  dvcnvrelem2  26252  dvcnvre  26253  ftc1lem3  26272  ftc1cn  26277  taylthlem1  26616  ulmdvlem3  26645  psercn  26669  abelth  26684  logcn  26892  cxpcn  26990  cxpcn2  26991  cxpcn3  26993  resqrtcn  26994  sqrtcn  26995  loglesqrt  27006  xrlimcnp  27213  efrlim  27214  ftalem3  27319  xrge0pluscn  34458  xrge0mulc1cn  34459  lmlimxrge0  34466  pnfneige0  34469  lmxrge0  34470  esumcvg  34604  cxpcncf1  35111  cvxpconn  35829  cvxsconn  35830  cvmsf1o  35859  cvmliftlem8  35879  cvmlift2lem9a  35890  cvmlift2lem11  35900  cvmlift3lem6  35911  ivthALT  36962  poimir  38410  broucube  38411  cnambfre  38425  ftc1cnnc  38449  areacirclem2  38466  areacirclem4  38468  fsumcncf  46714  ioccncflimc  46721  cncfuni  46722  icccncfext  46723  icocncflimc  46725  cncfiooicclem1  46729  cxpcncf2  46735  dvmptconst  46751  dvmptidg  46753  dvresntr  46754  itgsubsticclem  46811  dirkercncflem2  46940  dirkercncflem4  46942  fourierdlem32  46975  fourierdlem33  46976  fourierdlem62  47004  fourierdlem93  47035  fourierdlem101  47043