Theorem resttopon 21088

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 20841	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 472	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top)
3		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
4		toponmax 20853	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		ssexg 4912	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
6	3, 4, 5	syl2anr 496	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
7		resttop 21087	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
8	2, 6, 7	syl2anc 696	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
9		simpr 479	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10		sseqin2 3925	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
11	9, 10	sylib 208	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
12		simpl 474	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	4	adantr 472	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
14		elrestr 16212	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	12, 6, 13, 14	syl3anc 1439	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
16	11, 15	eqeltrrd 2804	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
17		elssuni 4575	. . . 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 16210	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20	6, 19	syldan 488	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
21		inss2 3942	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
22		vex 3307	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23	22	inex1 4907	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
24	23	elpw 4272	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
25	21, 24	mpbir 221	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
26	25	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
27		eqid 2724	. . . . . . 7 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))
28	26, 27	fmptd 6500	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
29		frn 6166	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴 → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
31	20, 30	eqsstrd 3745	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
32		sspwuni 4719	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
33	31, 32	sylib 208	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
34	18, 33	eqssd 3726	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
35		istopon 20840	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
36	8, 34, 35	sylanbrc 701	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596   ∈ wcel 2103  Vcvv 3304   ∩ cin 3679   ⊆ wss 3680  𝒫 cpw 4266  ∪ cuni 4544   ↦ cmpt 4837  ran crn 5219  ⟶wf 5997  ‘cfv 6001  (class class class)co 6765   ↾_t crest 16204  Topctop 20821  TopOnctopon 20838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-oadd 7684  df-er 7862  df-en 8073  df-fin 8076  df-fi 8433  df-rest 16206  df-topgen 16227  df-top 20822  df-topon 20839  df-bases 20873

This theorem is referenced by:  restuni  21089  stoig  21090  restsn2  21098  restlp  21110  restperf  21111  perfopn  21112  cnrest  21212  cnrest2  21213  cnrest2r  21214  cnpresti  21215  cnprest  21216  cnprest2  21217  restcnrm  21289  connsuba  21346  kgentopon  21464  1stckgenlem  21479  kgen2ss  21481  kgencn  21482  xkoinjcn  21613  qtoprest  21643  flimrest  21909  fclsrest  21950  flfcntr  21969  symgtgp  22027  dvrcn  22109  sszcld  22742  divcn  22793  cncfmptc  22836  cncfmptid  22837  cncfmpt2f  22839  cdivcncf  22842  cnmpt2pc  22849  icchmeo  22862  htpycc  22901  pcocn  22938  pcohtpylem  22940  pcopt  22943  pcopt2  22944  pcoass  22945  pcorevlem  22947  relcmpcmet  23236  limcvallem  23755  ellimc2  23761  limcres  23770  cnplimc  23771  cnlimc  23772  limccnp  23775  limccnp2  23776  dvbss  23785  perfdvf  23787  dvreslem  23793  dvres2lem  23794  dvcnp2  23803  dvcn  23804  dvaddbr  23821  dvmulbr  23822  dvcmulf  23828  dvmptres2  23845  dvmptcmul  23847  dvmptntr  23854  dvmptfsum  23858  dvcnvlem  23859  dvcnv  23860  lhop1lem  23896  lhop2  23898  lhop  23899  dvcnvrelem2  23901  dvcnvre  23902  ftc1lem3  23921  ftc1cn  23926  taylthlem1  24247  ulmdvlem3  24276  psercn  24300  abelth  24315  logcn  24513  cxpcn  24606  cxpcn2  24607  cxpcn3  24609  resqrtcn  24610  sqrtcn  24611  loglesqrt  24619  xrlimcnp  24815  efrlim  24816  ftalem3  24921  xrge0pluscn  30216  xrge0mulc1cn  30217  lmlimxrge0  30224  pnfneige0  30227  lmxrge0  30228  esumcvg  30378  cxpcncf1  30903  cvxpconn  31452  cvxsconn  31453  cvmsf1o  31482  cvmliftlem8  31502  cvmlift2lem9a  31513  cvmlift2lem11  31523  cvmlift3lem6  31534  ivthALT  32557  poimir  33674  broucube  33675  cnambfre  33690  ftc1cnnc  33716  areacirclem2  33733  areacirclem4  33735  fsumcncf  40511  ioccncflimc  40518  cncfuni  40519  icccncfext  40520  icocncflimc  40522  cncfiooicclem1  40526  cxpcncf2  40533  dvmptconst  40549  dvmptidg  40551  dvresntr  40552  itgsubsticclem  40611  dirkercncflem2  40741  dirkercncflem4  40743  fourierdlem32  40776  fourierdlem33  40777  fourierdlem62  40805  fourierdlem93  40836  fourierdlem101  40844