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Theorem resttopon 12267
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) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))

Proof of Theorem resttopon
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 topontop 12108 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 274 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
3 id 19 . . . 4 (𝐴𝑋𝐴𝑋)
4 toponmax 12119 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 ssexg 4037 . . . 4 ((𝐴𝑋𝑋𝐽) → 𝐴 ∈ V)
63, 4, 5syl2anr 288 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
7 resttop 12266 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ Top)
82, 6, 7syl2anc 408 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ Top)
9 simpr 109 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
10 sseqin2 3265 . . . . . 6 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
119, 10sylib 121 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑋𝐴) = 𝐴)
12 simpl 108 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
134adantr 274 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋𝐽)
14 elrestr 12055 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋𝐽) → (𝑋𝐴) ∈ (𝐽t 𝐴))
1512, 6, 13, 14syl3anc 1201 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑋𝐴) ∈ (𝐽t 𝐴))
1611, 15eqeltrrd 2195 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽t 𝐴))
17 elssuni 3734 . . . 4 (𝐴 ∈ (𝐽t 𝐴) → 𝐴 (𝐽t 𝐴))
1816, 17syl 14 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 (𝐽t 𝐴))
19 restval 12053 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
206, 19syldan 280 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
21 inss2 3267 . . . . . . . . 9 (𝑥𝐴) ⊆ 𝐴
22 vex 2663 . . . . . . . . . . 11 𝑥 ∈ V
2322inex1 4032 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2423elpw 3486 . . . . . . . . 9 ((𝑥𝐴) ∈ 𝒫 𝐴 ↔ (𝑥𝐴) ⊆ 𝐴)
2521, 24mpbir 145 . . . . . . . 8 (𝑥𝐴) ∈ 𝒫 𝐴
2625a1i 9 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ 𝒫 𝐴)
2726fmpttd 5543 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑥𝐽 ↦ (𝑥𝐴)):𝐽⟶𝒫 𝐴)
2827frnd 5252 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ⊆ 𝒫 𝐴)
2920, 28eqsstrd 3103 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ⊆ 𝒫 𝐴)
30 sspwuni 3867 . . . 4 ((𝐽t 𝐴) ⊆ 𝒫 𝐴 (𝐽t 𝐴) ⊆ 𝐴)
3129, 30sylib 121 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ⊆ 𝐴)
3218, 31eqssd 3084 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
33 istopon 12107 . 2 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽t 𝐴) ∈ Top ∧ 𝐴 = (𝐽t 𝐴)))
348, 32, 33sylanbrc 413 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  cin 3040  wss 3041  𝒫 cpw 3480   cuni 3706  cmpt 3959  ran crn 4510  cfv 5093  (class class class)co 5742  t crest 12047  Topctop 12091  TopOnctopon 12104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-rest 12049  df-topgen 12068  df-top 12092  df-topon 12105  df-bases 12137
This theorem is referenced by:  restuni  12268  stoig  12269  cnrest  12331  cnrest2  12332  cnrest2r  12333  cnptopresti  12334  cnptoprest  12335  cnptoprest2  12336  divcnap  12651  cncfmpt2fcntop  12681  cnplimcim  12732  cnlimcim  12736  cnlimc  12737  limccnpcntop  12740  limccnp2lem  12741  limccnp2cntop  12742  dvfvalap  12746  dvbss  12750  dvfgg  12753  dvcnp2cntop  12759  dvcn  12760  dvaddxxbr  12761  dvmulxxbr  12762
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