Theorem resttopon 20965

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 20718	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 481	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top)
3		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
4		toponmax 20730	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		ssexg 4804	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
6	3, 4, 5	syl2anr 495	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
7		resttop 20964	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
8	2, 6, 7	syl2anc 693	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
9		simpr 477	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
10		sseqin2 3817	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
11	9, 10	sylib 208	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
12		simpl 473	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	4	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
14		elrestr 16089	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	12, 6, 13, 14	syl3anc 1326	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
16	11, 15	eqeltrrd 2702	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
17		elssuni 4467	. . . 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 16087	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20	6, 19	syldan 487	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
21		inss2 3834	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
22		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23	22	inex1 4799	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
24	23	elpw 4164	. . . . . . . . 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 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))
28	26, 27	fmptd 6385	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
29		frn 6053	. . . . . 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 3639	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
32		sspwuni 4611	. . . 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 3620	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
35		istopon 20717	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
36	8, 34, 35	sylanbrc 698	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  restuni  20966  stoig  20967  restsn2  20975  restlp  20987  restperf  20988  perfopn  20989  cnrest  21089  cnrest2  21090  cnrest2r  21091  cnpresti  21092  cnprest  21093  cnprest2  21094  restcnrm  21166  connsuba  21223  kgentopon  21341  1stckgenlem  21356  kgen2ss  21358  kgencn  21359  xkoinjcn  21490  qtoprest  21520  flimrest  21787  fclsrest  21828  flfcntr  21847  symgtgp  21905  dvrcn  21987  sszcld  22620  divcn  22671  cncfmptc  22714  cncfmptid  22715  cncfmpt2f  22717  cdivcncf  22720  cnmpt2pc  22727  icchmeo  22740  htpycc  22779  pcocn  22817  pcohtpylem  22819  pcopt  22822  pcopt2  22823  pcoass  22824  pcorevlem  22826  relcmpcmet  23115  limcvallem  23635  ellimc2  23641  limcres  23650  cnplimc  23651  cnlimc  23652  limccnp  23655  limccnp2  23656  dvbss  23665  perfdvf  23667  dvreslem  23673  dvres2lem  23674  dvcnp2  23683  dvcn  23684  dvaddbr  23701  dvmulbr  23702  dvcmulf  23708  dvmptres2  23725  dvmptcmul  23727  dvmptntr  23734  dvmptfsum  23738  dvcnvlem  23739  dvcnv  23740  lhop1lem  23776  lhop2  23778  lhop  23779  dvcnvrelem2  23781  dvcnvre  23782  ftc1lem3  23801  ftc1cn  23806  taylthlem1  24127  ulmdvlem3  24156  psercn  24180  abelth  24195  logcn  24393  cxpcn  24486  cxpcn2  24487  cxpcn3  24489  resqrtcn  24490  sqrtcn  24491  loglesqrt  24499  xrlimcnp  24695  efrlim  24696  ftalem3  24801  xrge0pluscn  29986  xrge0mulc1cn  29987  lmlimxrge0  29994  pnfneige0  29997  lmxrge0  29998  esumcvg  30148  cxpcncf1  30673  cvxpconn  31224  cvxsconn  31225  cvmsf1o  31254  cvmliftlem8  31274  cvmlift2lem9a  31285  cvmlift2lem11  31295  cvmlift3lem6  31306  ivthALT  32330  poimir  33442  broucube  33443  cnambfre  33458  ftc1cnnc  33484  areacirclem2  33501  areacirclem4  33503  fsumcncf  40091  ioccncflimc  40098  cncfuni  40099  icccncfext  40100  icocncflimc  40102  cncfiooicclem1  40106  cxpcncf2  40113  dvmptconst  40129  dvmptidg  40131  dvresntr  40132  itgsubsticclem  40191  dirkercncflem2  40321  dirkercncflem4  40323  fourierdlem32  40356  fourierdlem33  40357  fourierdlem62  40385  fourierdlem93  40416  fourierdlem101  40424