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Theorem resttopon 12379
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 12220 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 274 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
3 id 19 . . . 4 (𝐴𝑋𝐴𝑋)
4 toponmax 12231 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 ssexg 4075 . . . 4 ((𝐴𝑋𝑋𝐽) → 𝐴 ∈ V)
63, 4, 5syl2anr 288 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
7 resttop 12378 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ Top)
82, 6, 7syl2anc 409 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ Top)
9 simpr 109 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
10 sseqin2 3300 . . . . . 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 12167 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋𝐽) → (𝑋𝐴) ∈ (𝐽t 𝐴))
1512, 6, 13, 14syl3anc 1217 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑋𝐴) ∈ (𝐽t 𝐴))
1611, 15eqeltrrd 2218 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽t 𝐴))
17 elssuni 3772 . . . 4 (𝐴 ∈ (𝐽t 𝐴) → 𝐴 (𝐽t 𝐴))
1816, 17syl 14 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 (𝐽t 𝐴))
19 restval 12165 . . . . . 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 3302 . . . . . . . . 9 (𝑥𝐴) ⊆ 𝐴
22 vex 2692 . . . . . . . . . . 11 𝑥 ∈ V
2322inex1 4070 . . . . . . . . . 10 (𝑥𝐴) ∈ V
2423elpw 3521 . . . . . . . . 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 5583 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑥𝐽 ↦ (𝑥𝐴)):𝐽⟶𝒫 𝐴)
2827frnd 5290 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ⊆ 𝒫 𝐴)
2920, 28eqsstrd 3138 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ⊆ 𝒫 𝐴)
30 sspwuni 3905 . . . 4 ((𝐽t 𝐴) ⊆ 𝒫 𝐴 (𝐽t 𝐴) ⊆ 𝐴)
3129, 30sylib 121 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ⊆ 𝐴)
3218, 31eqssd 3119 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
33 istopon 12219 . 2 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽t 𝐴) ∈ Top ∧ 𝐴 = (𝐽t 𝐴)))
348, 32, 33sylanbrc 414 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2689  cin 3075  wss 3076  𝒫 cpw 3515   cuni 3744  cmpt 3997  ran crn 4548  cfv 5131  (class class class)co 5782  t crest 12159  Topctop 12203  TopOnctopon 12216
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-rest 12161  df-topgen 12180  df-top 12204  df-topon 12217  df-bases 12249
This theorem is referenced by:  restuni  12380  stoig  12381  cnrest  12443  cnrest2  12444  cnrest2r  12445  cnptopresti  12446  cnptoprest  12447  cnptoprest2  12448  divcnap  12763  cncfmpt2fcntop  12793  cnplimcim  12844  cnlimcim  12848  cnlimc  12849  limccnpcntop  12852  limccnp2lem  12853  limccnp2cntop  12854  dvfvalap  12858  dvbss  12862  dvfgg  12865  dvcnp2cntop  12871  dvcn  12872  dvaddxxbr  12873  dvmulxxbr  12874
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