Theorem resttopon 23077

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 22829	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		id 22	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋)
3		toponmax 22842	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
4		ssexg 5263	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V)
5	2, 3, 4	syl2anr 597	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
6		resttop 23076	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top)
7	1, 5, 6	syl2an2r 685	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top)
8		simpr 484	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
9		sseqin2 4172	. . . . . 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 17334	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
14	11, 5, 12, 13	syl3anc 1373	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴))
15	10, 14	eqeltrrd 2834	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴))
16		elssuni 4889	. . . 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 17332	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
19	5, 18	syldan 591	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
20		inss2 4187	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
21		vex 3441	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
22	21	inex1 5257	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V
23	22	elpw 4553	. . . . . . . . 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 7054	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴)
27	26	frnd 6664	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴)
28	19, 27	eqsstrd 3965	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴)
29		sspwuni 5050	. . . 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 3948	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
32		istopon 22828	. 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴)))
33	7, 31, 32	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549  ∪ cuni 4858   ↦ cmpt 5174  ran crn 5620  ‘cfv 6486  (class class class)co 7352   ↾_t crest 17326  Topctop 22809  TopOnctopon 22826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-en 8876  df-fin 8879  df-fi 9302  df-rest 17328  df-topgen 17349  df-top 22810  df-topon 22827  df-bases 22862

This theorem is referenced by:  restuni  23078  stoig  23079  restsn2  23087  restlp  23099  restperf  23100  perfopn  23101  cnrest  23201  cnrest2  23202  cnrest2r  23203  cnpresti  23204  cnprest  23205  cnprest2  23206  restcnrm  23278  connsuba  23336  kgentopon  23454  1stckgenlem  23469  kgen2ss  23471  kgencn  23472  xkoinjcn  23603  qtoprest  23633  flimrest  23899  fclsrest  23940  flfcntr  23959  efmndtmd  24017  symgtgp  24022  dvrcn  24100  sszcld  24734  divcnOLD  24785  divcn  24787  cncfmptc  24833  cncfmptid  24834  cncfmpt2f  24836  cdivcncf  24842  cnmpopc  24850  icchmeo  24866  icchmeoOLD  24867  htpycc  24907  pcocn  24945  pcohtpylem  24947  pcopt  24950  pcopt2  24951  pcoass  24952  pcorevlem  24954  relcmpcmet  25246  mulcncf  25374  limcvallem  25800  ellimc2  25806  limcres  25815  cnplimc  25816  cnlimc  25817  limccnp  25820  limccnp2  25821  dvbss  25830  perfdvf  25832  dvreslem  25838  dvres2lem  25839  dvcnp2  25849  dvcnp2OLD  25850  dvcn  25851  dvaddbr  25868  dvmulbr  25869  dvmulbrOLD  25870  dvcmulf  25876  dvmptres2  25894  dvmptcmul  25896  dvmptntr  25903  dvmptfsum  25907  dvcnvlem  25908  dvcnv  25909  lhop1lem  25946  lhop2  25948  lhop  25949  dvcnvrelem2  25951  dvcnvre  25952  ftc1lem3  25973  ftc1cn  25978  taylthlem1  26309  ulmdvlem3  26339  psercn  26364  abelth  26379  logcn  26584  cxpcn  26682  cxpcnOLD  26683  cxpcn2  26684  cxpcn3  26686  resqrtcn  26687  sqrtcn  26688  loglesqrt  26699  xrlimcnp  26906  efrlim  26907  efrlimOLD  26908  ftalem3  27013  xrge0pluscn  33974  xrge0mulc1cn  33975  lmlimxrge0  33982  pnfneige0  33985  lmxrge0  33986  esumcvg  34120  cxpcncf1  34629  cvxpconn  35307  cvxsconn  35308  cvmsf1o  35337  cvmliftlem8  35357  cvmlift2lem9a  35368  cvmlift2lem11  35378  cvmlift3lem6  35389  ivthALT  36400  poimir  37713  broucube  37714  cnambfre  37728  ftc1cnnc  37752  areacirclem2  37769  areacirclem4  37771  fsumcncf  46000  ioccncflimc  46007  cncfuni  46008  icccncfext  46009  icocncflimc  46011  cncfiooicclem1  46015  cxpcncf2  46021  dvmptconst  46037  dvmptidg  46039  dvresntr  46040  itgsubsticclem  46097  dirkercncflem2  46226  dirkercncflem4  46228  fourierdlem32  46261  fourierdlem33  46262  fourierdlem62  46290  fourierdlem93  46321  fourierdlem101  46329