Theorem resttopon 23190

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 22940	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 22953	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5341	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 596	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23189	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 684	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		simpr 484	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
9		sseqin2 4244	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴)
10	8, 9	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴)
11		simpl 482	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
12	3	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽)
13		elrestr 17488	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	11, 5, 12, 13	syl3anc 1371	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	10, 14	eqeltrrd 2845	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
16		elssuni 4961	. . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
17	15, 16	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴))
18		restval 17486	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19	5, 18	syldan 590	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20		inss2 4259	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
21		vex 3492	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
22	21	inex1 5335	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
23	22	elpw 4626	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴)
24	20, 23	mpbir 231	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴
25	24	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴)
26	25	fmpttd 7149	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
27	26	frnd 6755	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
28	19, 27	eqsstrd 4047	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
29		sspwuni 5123	. . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
30	28, 29	sylib 218	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴)
31	17, 30	eqssd 4026	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
32		istopon 22939	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
33	7, 31, 32	sylanbrc 582	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-en 9004  df-fin 9007  df-fi 9480  df-rest 17482  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974

This theorem is referenced by:  restuni  23191  stoig  23192  restsn2  23200  restlp  23212  restperf  23213  perfopn  23214  cnrest  23314  cnrest2  23315  cnrest2r  23316  cnpresti  23317  cnprest  23318  cnprest2  23319  restcnrm  23391  connsuba  23449  kgentopon  23567  1stckgenlem  23582  kgen2ss  23584  kgencn  23585  xkoinjcn  23716  qtoprest  23746  flimrest  24012  fclsrest  24053  flfcntr  24072  efmndtmd  24130  symgtgp  24135  dvrcn  24213  sszcld  24858  divcnOLD  24909  divcn  24911  cncfmptc  24957  cncfmptid  24958  cncfmpt2f  24960  cdivcncf  24966  cnmpopc  24974  icchmeo  24990  icchmeoOLD  24991  htpycc  25031  pcocn  25069  pcohtpylem  25071  pcopt  25074  pcopt2  25075  pcoass  25076  pcorevlem  25078  relcmpcmet  25371  mulcncf  25499  limcvallem  25926  ellimc2  25932  limcres  25941  cnplimc  25942  cnlimc  25943  limccnp  25946  limccnp2  25947  dvbss  25956  perfdvf  25958  dvreslem  25964  dvres2lem  25965  dvcnp2  25975  dvcnp2OLD  25976  dvcn  25977  dvaddbr  25994  dvmulbr  25995  dvmulbrOLD  25996  dvcmulf  26002  dvmptres2  26020  dvmptcmul  26022  dvmptntr  26029  dvmptfsum  26033  dvcnvlem  26034  dvcnv  26035  lhop1lem  26072  lhop2  26074  lhop  26075  dvcnvrelem2  26077  dvcnvre  26078  ftc1lem3  26099  ftc1cn  26104  taylthlem1  26433  ulmdvlem3  26463  psercn  26488  abelth  26503  logcn  26707  cxpcn  26805  cxpcnOLD  26806  cxpcn2  26807  cxpcn3  26809  resqrtcn  26810  sqrtcn  26811  loglesqrt  26822  xrlimcnp  27029  efrlim  27030  efrlimOLD  27031  ftalem3  27136  xrge0pluscn  33886  xrge0mulc1cn  33887  lmlimxrge0  33894  pnfneige0  33897  lmxrge0  33898  esumcvg  34050  cxpcncf1  34572  cvxpconn  35210  cvxsconn  35211  cvmsf1o  35240  cvmliftlem8  35260  cvmlift2lem9a  35271  cvmlift2lem11  35281  cvmlift3lem6  35292  ivthALT  36301  poimir  37613  broucube  37614  cnambfre  37628  ftc1cnnc  37652  areacirclem2  37669  areacirclem4  37671  fsumcncf  45799  ioccncflimc  45806  cncfuni  45807  icccncfext  45808  icocncflimc  45810  cncfiooicclem1  45814  cxpcncf2  45820  dvmptconst  45836  dvmptidg  45838  dvresntr  45839  itgsubsticclem  45896  dirkercncflem2  46025  dirkercncflem4  46027  fourierdlem32  46060  fourierdlem33  46061  fourierdlem62  46089  fourierdlem93  46120  fourierdlem101  46128