Theorem resttopon 21200

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 20952	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 468	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top)
3		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
4		toponmax 20965	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		ssexg 5012	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
6	3, 4, 5	syl2anr 586	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
7		resttop 21199	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
8	2, 6, 7	syl2anc 575	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
9		simpr 473	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10		sseqin2 4027	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
11	9, 10	sylib 209	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
12		simpl 470	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	4	adantr 468	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
14		elrestr 16314	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	12, 6, 13, 14	syl3anc 1483	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
16	11, 15	eqeltrrd 2897	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
17		elssuni 4672	. . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
18	16, 17	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
19		restval 16312	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20	6, 19	syldan 581	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
21		inss2 4041	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
22		vex 3405	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23	22	inex1 5007	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
24	23	elpw 4368	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
25	21, 24	mpbir 222	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
26	25	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
27	26	fmpttd 6617	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
28	27	frnd 6273	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
29	20, 28	eqsstrd 3847	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
30		sspwuni 4814	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
31	29, 30	sylib 209	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
32	18, 31	eqssd 3826	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
33		istopon 20951	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
34	8, 32, 33	sylanbrc 574	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2157  Vcvv 3402   ∩ cin 3779   ⊆ wss 3780  𝒫 cpw 4362  ∪ cuni 4641   ↦ cmpt 4934  ran crn 5325  ‘cfv 6111  (class class class)co 6884   ↾_t crest 16306  Topctop 20932  TopOnctopon 20949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-oadd 7810  df-er 7989  df-en 8203  df-fin 8206  df-fi 8566  df-rest 16308  df-topgen 16329  df-top 20933  df-topon 20950  df-bases 20985

This theorem is referenced by:  restuni  21201  stoig  21202  restsn2  21210  restlp  21222  restperf  21223  perfopn  21224  cnrest  21324  cnrest2  21325  cnrest2r  21326  cnpresti  21327  cnprest  21328  cnprest2  21329  restcnrm  21401  connsuba  21458  kgentopon  21576  1stckgenlem  21591  kgen2ss  21593  kgencn  21594  xkoinjcn  21725  qtoprest  21755  flimrest  22021  fclsrest  22062  flfcntr  22081  symgtgp  22139  dvrcn  22221  sszcld  22854  divcn  22905  cncfmptc  22948  cncfmptid  22949  cncfmpt2f  22951  cdivcncf  22954  cnmpt2pc  22961  icchmeo  22974  htpycc  23013  pcocn  23050  pcohtpylem  23052  pcopt  23055  pcopt2  23056  pcoass  23057  pcorevlem  23059  relcmpcmet  23349  limcvallem  23872  ellimc2  23878  limcres  23887  cnplimc  23888  cnlimc  23889  limccnp  23892  limccnp2  23893  dvbss  23902  perfdvf  23904  dvreslem  23910  dvres2lem  23911  dvcnp2  23920  dvcn  23921  dvaddbr  23938  dvmulbr  23939  dvcmulf  23945  dvmptres2  23962  dvmptcmul  23964  dvmptntr  23971  dvmptfsum  23975  dvcnvlem  23976  dvcnv  23977  lhop1lem  24013  lhop2  24015  lhop  24016  dvcnvrelem2  24018  dvcnvre  24019  ftc1lem3  24038  ftc1cn  24043  taylthlem1  24364  ulmdvlem3  24393  psercn  24417  abelth  24432  logcn  24630  cxpcn  24723  cxpcn2  24724  cxpcn3  24726  resqrtcn  24727  sqrtcn  24728  loglesqrt  24736  xrlimcnp  24932  efrlim  24933  ftalem3  25038  xrge0pluscn  30334  xrge0mulc1cn  30335  lmlimxrge0  30342  pnfneige0  30345  lmxrge0  30346  esumcvg  30496  cxpcncf1  31021  cvxpconn  31569  cvxsconn  31570  cvmsf1o  31599  cvmliftlem8  31619  cvmlift2lem9a  31630  cvmlift2lem11  31640  cvmlift3lem6  31651  ivthALT  32673  poimir  33774  broucube  33775  cnambfre  33789  ftc1cnnc  33815  areacirclem2  33832  areacirclem4  33834  fsumcncf  40589  ioccncflimc  40596  cncfuni  40597  icccncfext  40598  icocncflimc  40600  cncfiooicclem1  40604  cxpcncf2  40611  dvmptconst  40627  dvmptidg  40629  dvresntr  40630  itgsubsticclem  40688  dirkercncflem2  40818  dirkercncflem4  40820  fourierdlem32  40853  fourierdlem33  40854  fourierdlem62  40882  fourierdlem93  40913  fourierdlem101  40921