Theorem resttopon 23459

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 23211	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 23224	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5281	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 609	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23458	. . 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 4169	. . . . . 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 17579	. . . . . 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 2862	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
15		elssuni 4899	. . . 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 17577	. . . . . 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 4183	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
20		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
21	20	inex1 5277	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
22	21	elpw 4561	. . . . . . . . 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 7107	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
26	25	frnd 6710	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
27	18, 26	eqsstrd 3965	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
28		sspwuni 5060	. . . 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 3948	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
31		istopon 23210	. 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 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5652  ‘cfv 6531  (class class class)co 7412   ↾_t crest 17571  Topctop 23191  TopOnctopon 23208

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-en 8958  df-fin 8961  df-fi 9387  df-rest 17573  df-topgen 17594  df-top 23192  df-topon 23209  df-bases 23244

This theorem is used by:  restuni  23460  stoig  23461  restsn2  23469  restlp  23481  restperf  23482  perfopn  23483  cnrest  23583  cnrest2  23584  cnrest2r  23585  cnpresti  23586  cnprest  23587  cnprest2  23588  restcnrm  23660  connsuba  23718  kgentopon  23837  1stckgenlem  23852  kgen2ss  23854  kgencn  23855  xkoinjcn  23986  qtoprest  24016  flimrest  24282  fclsrest  24323  flfcntr  24342  efmndtmd  24400  symgtgp  24405  dvrcn  24483  sszcld  25117  divcn  25169  cncfmptc  25213  cncfmptid  25214  cncfmpt2f  25216  cdivcncf  25222  cnmpopc  25229  icchmeo  25242  htpycc  25281  pcocn  25318  pcohtpylem  25320  pcopt  25323  pcopt2  25324  pcoass  25325  pcorevlem  25327  relcmpcmet  25619  mulcncf  25747  limcvallem  26171  ellimc2  26177  limcres  26186  cnplimc  26187  cnlimc  26188  limccnp  26191  limccnp2  26192  dvbss  26201  perfdvf  26203  dvreslem  26209  dvres2lem  26210  dvcnp2  26220  dvcn  26221  dvaddbr  26238  dvmulbr  26239  dvcmulf  26245  dvmptres2  26262  dvmptcmul  26264  dvmptntr  26271  dvmptfsum  26275  dvcnvlem  26276  dvcnv  26277  lhop1lem  26313  lhop2  26315  lhop  26316  dvcnvrelem2  26318  dvcnvre  26319  ftc1lem3  26338  ftc1cn  26343  taylthlem1  26682  ulmdvlem3  26711  psercn  26735  abelth  26750  logcn  26957  cxpcn  27055  cxpcn2  27056  cxpcn3  27058  resqrtcn  27059  sqrtcn  27060  loglesqrt  27071  xrlimcnp  27278  efrlim  27279  ftalem3  27384  xrge0pluscn  34554  xrge0mulc1cn  34555  lmlimxrge0  34562  pnfneige0  34565  lmxrge0  34566  esumcvg  34700  cxpcncf1  35207  cvxpconn  35976  cvxsconn  35977  cvmsf1o  36006  cvmliftlem8  36026  cvmlift2lem9a  36037  cvmlift2lem11  36047  cvmlift3lem6  36058  ivthALT  37093  poimir  38539  broucube  38540  cnambfre  38554  ftc1cnnc  38578  areacirclem2  38595  areacirclem4  38597  fsumcncf  46832  ioccncflimc  46839  cncfuni  46840  icccncfext  46841  icocncflimc  46843  cncfiooicclem1  46847  cxpcncf2  46853  dvmptconst  46869  dvmptidg  46871  dvresntr  46872  itgsubsticclem  46929  dirkercncflem2  47058  dirkercncflem4  47060  fourierdlem32  47093  fourierdlem33  47094  fourierdlem62  47122  fourierdlem93  47153  fourierdlem101  47161